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\begin{document}
\title{Dependence on state preparation of noise induced effects in multiqubit systems }
\author{Athanasios C. Tzemos, Demetris P.K. Ghikas\footnote{Corresponding Author, Tel: +302610-997460, FAX: +302610-997617}}
\address{$�$ Department of Physics, University of Patras, Patras 26500, Greece}
\ead{[email protected], [email protected]}
\begin{abstract}
The perturbation of multiqubit systems by an external noise can induce various effects like decoherence, stochastic resonance and anti-resonance, and noise-shielding. We investigate how the appearance of these effects on disentanglement time depends on the initial preparation of the systems. We present results for $2$, $3$ and $4$-qubit chains in various arrangements and observe a clear de\-pen\-dence on the combination of initial geometry of the state space and the placement of noise. Finally, we see that temperature can play a constructive role for the control of these noise induced effects.
\end{abstract}
\pacs{03.67.Bg, 03.67.Mn, 05.10.Gg}
\submitto{\JPA}
\maketitle
\section{Introduction}
Multiqubit systems are the building blocks for the architecture of quantum information storing and processing \cite{Nielsen}. Their efficiency depends strongly on the possibility to retain quantum coherence. But these quantum systems are physically embedded and constantly interacting with their local quantum and classical environments which affect coherence \cite{Schlosshauer}. Thus being fundamentally open quantum systems, they dissipate and decohere with time constants that depend on their parameters and coupling constants and on the strength of perturbation by external agents. There have been many efforts to increase the decoherence time, either passively by increasing isolation or engineering the existence of decoherence free subspaces \cite{Lidar1} or actively by affecting the dynamics with external controls \cite{Lidar2,Knill,Franco}.
As it has turned out, a counterintuitive and unexpected external perturbation with a positive role is the action by a noise source, quantum or classical. Normally one expects that noise would only increase decoherence in a monotonous way, namely increased destruction of coherence with higher levels of noise. But it has been found that noise may be used to isolate a quantum system, the noise-shielding effect (N.S.), and may influence positively or negatively the decoherence time in a monotonous or non-monotonous way. These are the quantum Zeno effects and the stochastic resonance (S.R.) or stochastic anti-resonance (S.AR.) effects \cite{Misra,Zaffino,Francica,Ando,Rivas,Wellens,Ghikas}. But it appears that the manifestation of these effects depends, apart from the detailed dynamical setup, on the way the system has been prepared.
From our preliminary investigation of noise effects on two-qubit systems \cite{Ghikas}, it became evident that certain classes of initial states are more sensitive than others. Considering this not to be accidental, we extended our study of the state preparation dependence of the noisy perturbations, to $3$ and $4$-qubit Heisenberg XY chains. We have found a clear correlation between the way a quantum system of qubits reacts to an external classical noise and the geometry of the initial states. As a tool we used a Master equation for modeling the dissipative and decoherence processes, and applied an external classical gaussian white noise as a stochastic control. Our investigation is based on the decoherence properties of a 2-qubit subsystem. In the case of two qubits we have only the bosonic environment and the external classical noise. In the cases of $3$ and $4$-qubit systems we consider the extra qubits, after they are traced out, as a form of local fermionic environment. We quantify the entanglement with the concurrence of the 2-qubit subsystem. We study numerically the dependence of disentanglement time (or entanglement sudden death (E.S.D.)) \cite{Yu1,Yu2,Yu3,Yu4,Yu5,Yu6,Cunha} on the strength of the applied noise for zero and nonzero temperature and for various initial states. We have observed the effects of increase of decoherence, stochastic resonance, stochastic anti-resonance and noise shielding.
The main result is that these effects depend strongly on the initial preparation of the compound system and the placement of classical noise. The importance of this result could be appreciated in the cases where the behavior of a studied subsystem depends on the preparation of a bigger system whose parts have been traced out, but they are in interaction with the subsystem of interest. Our results are summarized as follows:
\begin {itemize}\item{ 2 qubits
\begin{enumerate}
\item{Stochastic anti-resonance is observed if the initial density matrix
contains population elements $\rho_{22}$ or $\rho_{33}$. This becomes evident in Bell states $|\Psi\rangle$, (\fref{2q1} (a) (upper curve)), while the entanglement of Bell states $|\Phi\rangle$ decays monotonically over increasing noise strength (\fref{2q1} (b))}.
\item{For vanishing dissipation rate ($\gamma\to 0$), the Hilbert subspace which exhibits stochastic anti-resonance tends to become a decoherence free subspace \cite{Ghikas}.}
\item{There are no noise shields in the 2-qubit case. Noise shields require the exten\-sion of the system to more qubits, in order to apply noise on the local fermionic environment.}
\item{Temperature degrades stochastic anti-resonance very quickly. An average exci\-ta\-tion number $\langle n\rangle\simeq0.5$ is enough to eliminate the appearance of the anti-resonance (\fref{2q1} (a) (lower curve)).}
\end{enumerate}}
\item{ 3 qubits
\begin{enumerate}
\item{Multiple resonances (stochastic resonance and stochastic anti-resonance) are observed in two product states when noise affects the traced out qubit (\fref{3q} (a)). Moreover, there are two product states that exhibit N.S. behavior , and two that exhibit S.AR. (\ref{a3q1}).}
\item{Product states with parallel spins do not present interesting behavior. Noise decreases monotonically their entanglement evolution. This holds for any ar\-ran\-gement of noise perturbation (\ref{a3q1}).}
\item{By altering the initial 3-qubit preparation of a given 2-qubit state, we observe different behaviour in the 2-qubit disentanglement time. This becomes evident with the reduced 2-qubit state of a W state. A small change of initial 3-qubit pre\-paration results in N.S., while W state preparation results in S.AR. (\fref{3q2} (a,b)).}
\item{Bell state $|\Phi\rangle$ preparations exhibit N.S. when noise affects the traced out qubit, for all of the initial 3-qubit preparations (\fref{3q} (b) and \ref{a3q2}).}
\item{Most of the Bell state $|\Psi\rangle$ preparations exhibit N.S. when noise affects the traced out qubit, but there is an initial system preparation which results in S.R. (\fref{3q}(c) and \ref{a3q3}).}
\end{enumerate}}
\item{ 4 qubits
\begin{enumerate}
\item{Product states with parallel spins exhibit the same behavior as in the 3-qubit case (\ref{a4q1}).}
\item{Most of the product states exhibit noise shield when noise is environmental (\fref{4q1} (a) and \ref{a4q1}).}
\item{There are four product states which exhibit different kinds of effects, depending on the placement of noise and one that exhibits S.AR. behavior when noise is environmental (\ref{a4q1}).}
\item{The noise shields tend to become weaker if we increase the anisotropy of the system (\fref{4q1} (b)).}
\item{Most of the Bell state $|\Phi^+\rangle,|\Psi^+\rangle$ preparations continue to exhibit N.S. when noise is environmental, except from two $|\Phi^+\rangle$ preparations that result in stochastic anti-resonance, two preparations of $|\Psi^+\rangle$ which result in S.A.R., one $|\Psi^+\rangle$ preparation that exhibits multiple resonances and one $|\Psi^+\rangle$ preparation that exhibits S.R. (the last one when noise is internal). (\fref{4q12}(a) and \ref{a4q2}, \ref{a4q3}).}
\item{The noise shields become stronger by applying noise in both of the traced out qubits rather than one of them (\fref{4q12} (b,c)). Temperature degrades their magnitude, but does not change their shape. There is increase of to\-lerance against temperature compared with the 2-qubit case, because of the environmental qubits. We can observe a clear N.S. behavior with excitation values up to $\langle n\rangle\simeq6$ (\fref{4q3}(a)).}
\item{Even though it is difficult to determine the aforementioned initial preparations which exhibit S.AR, we can control this behaviour by increasing temperature. Above a critical value of the latter, S.AR. disappears and we observe N.S.. This attributes an interesting positive role to the temperature, for the control of entanglement evolution (\fref{4q3}(b)).}
\end{enumerate}}
\end{itemize}
A complete list of the effects is presented in the tables of Appendix A and B. In the final paragraph we classify our results in terms of the observed effects and make comments for the interplay between the geometry of the initial system preparation and the placement of noise, which seems to be crucial for their appearance.
\section{XY Heisenberg model}
The general form of a N-spin XY chain (for spin $1/2$ particles) with nearest-neighbor interaction is
\begin{eqnarray}
H_{XY}=\sum_{n=1}^N\Big(J_xS_n^xS_{n+1}^x+J_yS_n^yS_{n+1}^y\Big),
\end{eqnarray}
with $S_n^i=\frac{1}{2}\sigma^i_n (i=x,y,z)$ the spin-$1/2$ operators, $\sigma_n^i$ the corresponding Pauli's ope\-rators, $\hbar=1$ and $S_{N+1}=S_1$ (periodic boundary condition). The chain is ferromagnetic if $J_i<0$, and anti-ferromagnetic if $J_i>0$. Here we study this chain in the presence of a constant external magnetic field $\omega_0$ along the z-axis. Consequently, the unperturbed system Hamiltonian is:
\begin{eqnarray}
H_{0}=H_{XY}+\sum_{n=1}^N\omega_{0} S^z_n
\label{Hamiltonian},
\end{eqnarray}
for $N=3,4$.
The XY model has been studied for many decades because of its interesting and unusual features \cite{Wang}. It is an example of integrable system. While the iso\-tro\-pic Heisenberg chain is solved with the use of Bethe Ansatz, the XY model is solved by means of Jordan-Wigner transformation \cite{Jordan} which was introduced in $1928$ and applied to it by Lieb \etal \cite{Lieb} in $1961$. For more information see \cite{Parkinson}. Furthermore, due to its mathematical simplicity, it is suitable for our purpose and provides a basic unit for many quantum information implementations, such as N.M.R. quantum computation and quantum teleportation \cite{Rao,Yeo}.
\section{Markovian Master Equation}
We employ a Markovian form of the Master equation, where the classical gaussian white noise $\xi(t)$ affects the magnetic field and is appropriately incorporated in the double commutator \cite{Luczka}. For a thermal bath with $T\geq0$ the Master equation is
\begin{eqnarray}
\frac{d\rho_s}{dt}=&\nonumber-i[H_0,\rho_s]+\gamma(\langle n\rangle+1)\sum_{n=1}^N\Big(D[S^-_n]\rho_s\Big)\\&+\gamma \langle n\rangle\sum_{n=1}^N\Big(D[S^+_n]\rho_s\Big)-M_z[V_z,[V_z,\rho_s]],\label{master}
\end{eqnarray}
where $V_z$ is the spin operator affected by noise. It can be any of $S_n^z$ or sum of these. $M_z$ is the corresponding noise parameter. For example $M_4$ means that $V_4=S_4^z$, while $M_{34}$ means that $V_{34}=S_3^z+S_4^z$. Furthermore $D[S^-_n]\rho_s=S^-_n\rho_s(S^-_n)^{\dagger}-\{(S^-_n)^{\dagger}S^-_n,\rho_s\}/2$. The coefficient $\gamma$ is the rate of population relaxation and $\langle n\rangle$ denotes the average excitation quanta of the bath. It depends monotonically on the temperature and it is used to parameterize the latter. For $T=0$ we have $ \langle n\rangle=0$ and for
$T\rightarrow\infty$, $ \langle n\rangle\rightarrow\infty$.
The first term describes the unitary evolution of the system, the second and third terms the interaction between the system and the thermal bath, while the last one is the addition of an external classical gaussian white noise \cite{Luczka}.
We assumed that each qubit has the same interaction with the environment, something that implies constraints on the value of coupling constants \cite{Ghikas,Wang}.
The eigen\-values of the 3-qubit Hamiltonian are:
\begin{eqnarray}\frac{{J+\omega\pm\sqrt{(J-2\omega)^2+3\Delta^2}}}{2}\\
\frac{{J-\omega\pm\sqrt{(J+2\omega)^2+3\Delta^2}}}{2}\\
\frac{-J\pm{\omega}}{2},\,\,\textnormal{double eigenvalues},
\end{eqnarray}
while the eigenvalues of the 4-qubit Hamiltonian are:
\begin{eqnarray}\pm\sqrt{{J}^{2}+{\Delta}^{2}+2{\omega}^{2}+\sqrt {{\Delta}^{4}+\left( 4{
\omega}^{2}+2{J}^{2} \right) {\Delta}^{2}+ \left( {J}^{2}-2{\omega}^{2}\right) ^{2}}}\\
\pm\sqrt{{J}^{2}+{\Delta}^{2}+2{\omega}^{2}-\sqrt {{\Delta}^{4}+\left( 4{
\omega}^{2}+2{J}^{2} \right) {\Delta}^{2}+ \left( {J}^{2}-2{\omega}^{2}\right) ^{2}}}\\
-J\pm\sqrt{\omega^2+\Delta^2},\\
J\pm\sqrt{\omega^2+\Delta^2}\\
\pm\omega\,\,\textnormal{double eigenvalues}\\
0\,\,\textnormal{fourfold eigenvalue}
\end{eqnarray}
where $J=\frac{J_x+J_y}{2}$ and $\Delta=\frac{J_x-J_y}{2}$.
Assuming common $\gamma$ for all of the qubits, we work with $\omega=4$, $J=0.2$, $\Delta=0.1$, so that the interaction between them has not altered energy level separations by more than $10\%$ from that of the non-interacting system. Moreover, we choose $\gamma=0.01$ for the weak coupling approximation ($\omega\gg\gamma$) to remain valid.
\begin{figure}
\caption{The two different arrangements of the unperturbed system: $2$ qubit subsystem of a $3$-qubit chain (left) and a $4$-qubit chain (right) inside a common bosonic heat bath. The subsystem (green circles) interacts with the bosonic bath and the local fermionic environment (yellow circles). The main difference between them, is that in the case of $3$ qubits all of them interact directly, while in the case of $4$ qubits the interaction between $1-3$ and $2-4$ is indirect.}
\label{a1}
\end{figure}
\section{Entanglement Evolution}
Entanglement is a fundamental property of quantum systems and lies in the heart of quantum information and computation theory \cite{Nielsen,Horodecki}. One of the most difficult problems in contemporary information theory is to construct measures for the quantification of quantum entanglement. While the entanglement of a bipartite system is well understood, many measures have been presented for multipartite systems and most of them are related to pure states. This makes even more difficult the study of decohering systems, where initially pure states tend to become mixed, because of the constant interaction with their environment. That is why we choose to work with the bipartite entanglement, which is quantified by concurrence and gives us the advantage to describe both pure and mixed states \cite{Horodecki2,Kus,Wooters,Wooters2}.
For a system described by the density matrix $\rho$, the concurrence $C$ is
\begin{eqnarray}
C=\max(\sqrt{\lambda_1}-\sqrt{\lambda_2}-\sqrt{\lambda_3}-\sqrt{\lambda_4},0),
\end{eqnarray}
where $\lambda_1,\lambda_2,\lambda_3,\lambda_4$ are the eigenvalues of spin flipped density matrix R (with $\lambda_1$ the largest one), the definition of which is:
\begin{eqnarray}
R=\rho(\sigma_y\otimes \sigma_y)\rho^*(\sigma_y\otimes \sigma_y).
\end{eqnarray}
$C$ lies in the range $[0,1]$. $C=0$ corresponds to a product state, while $C=1$ to a maximally entangled state. All of the other states inside this range are {\itshape partially entangled\/}.
The basis states for our system are formed by the tensor product of $S_n^z$ eigenstates $\{|e\rangle_n,|g\rangle_n\}$. In the 3-qubit case we have
\begin{eqnarray}
\{|eee\rangle,|eeg\rangle,|ege\rangle,|egg\rangle,|gee\rangle,|geg\rangle,|gge\rangle,|ggg\rangle\}
\end{eqnarray}
while in 4-qubit case
\begin{eqnarray}
\{|eeee\rangle,|eeeg\rangle,|eege\rangle,|eegg\rangle,|egee\rangle,|egeg\rangle,|egge\rangle,|eggg\rangle,\\|geee\rangle,
|geeg\rangle,|gege\rangle,|gegg\rangle,|ggee\rangle,|ggeg\rangle,|ggge\rangle,|gggg\rangle\}.
\end{eqnarray}
A simplifying feature of our Hamiltonian is that the dynamics of system splits into two independently evolving sets of equations, resulting in smaller density matrices and consequently smaller differential systems. For three qubits case we get the density matrix \ref{d3} and for four qubits the density matrix \ref{d4} correspondingly. They both have the half the number of elements of the original density matrix and result in the following reduced density matrix:
\begin{eqnarray}\rho^{r}=\left[ \begin{array}{cccccc}
\rho_{1,1}^{r} & & &\rho_{1,4}^{r} \\
&\rho_{2,2}^{r} &\rho_{2,3}^{r} & \\
&\rho_{3,2}^{r} &\rho_{3,3}^{r} & \\
\rho_{4,1}^{r} & & &\rho_{4,4}^{r}
\end{array}\right],\textrm{where blank entries are equal to $0$.}\end{eqnarray}
The mixed state defined by this submatrix is called X state (non-zero elements along the diagonal and the anti-diagonal) and arises in many physical situations \cite{Rau,Rau2,James}. It has the advantage that includes Bell states $|\Psi^{\pm}\rangle, |\Phi^{\pm}\rangle$, product states and mixed states. The concurrence of this matrix is equal to:
\begin{eqnarray}
C=\max\{0,C_1,C_2\},
\end{eqnarray}
where
\begin{eqnarray}
&C_1=2(|\rho_{4,1}^{r}|-\sqrt{\rho_{3,3}^{r}\rho_{2,2}^{r}}),\\&
C_2=2(|\rho_{3,2}^{r}|-\sqrt{\rho_{4,4}^{r}\rho_{1,1}^{r}}),
\end{eqnarray}
It is evident that the reduced matrix for 2 qubits will inherit elements from the full system matrix which is either $8\times8$ (3-qubits case) or $16\times16$ (4-qubits case). This means that different initial preparations of the compound system can lead to the same reduced matrix form, something expected, but with great consequences for the entanglement evolution, as we will show in the next sections.
\section{2-qubit chain}
For the sake of completeness we provide two new diagrams related to those of \cite{Ghikas}. In the 2-qubit chain there is no ambiguity for the initial state preparation, since we do not trace out any qubit (we want to study entanglement). The system decoheres in the presence of a bosonic bath and external classical white noise affecting the magnetic field. What we found is that if the initial preparation of qubits contains the diagonal matrix elements $\rho_{22}$ or $\rho_{33}$, then the entanglement evolves non-monotonically over noise. There is a value of noise which causes the shortest disentanglement time and has to be avoided. This is what we call stochastic anti-resonance (S.AR.). The effect becomes evident in the Bell states $|\Psi\rangle$ (\fref{2q1} (a) (upper curve)). Bell states $|\Phi\rangle$ decay monotonically, since their evolution begins outside the critical subspace (\fref{2q1} (b)). Regarding the bath temperature, it plays a negative role for the S.AR., something normally expected. A small increase of temperature is enough to eliminate the shape of S.AR. (\fref{2q1} (a) (lower curve)).
\begin{figure}
\caption{Entanglement sudden death time (E.S.D.) for $2$ interacting qubits, in the presence of a bosonic environment and external classical white noise. (a) Bell state $|\Psi^+\rangle$. The upper curve corresponds to zero temperature. There is a clear anti-resonance behaviour. The lower curve shows that a small increase of temperature erases the anti-resonance. (b) Bell state $|\Phi^+\rangle$ for $T=0$. There is no anti-resonance in this case. This shows that the geometry of the initial state is important for the appearance of this effect. Noise is collective.
($\omega = 1, \gamma = 0.01, J = \Delta = 0.1$ )}
\label{2q1}
\end{figure}
\section{3-qubit chain with $T=0$}
The reduced density matrix has the general form:
\begin{eqnarray}\label{3q1}
\rho^{r}=\left[ \begin{array}{cccccc}
\sum_{i=1}^{2}\rho_{i,i}&0&0&\sum_{i=1}^{2}\rho_{i,i+6}\\ 0&\sum_{i=3}^{4}\rho_{i,i}&\sum_{i=3}^{4}\rho_{i,i+2}&0\\0&\sum_{i=3}^{4}\rho_{i+2,i}& \sum_{i=5}^{6}\rho_{i,i}&0\\ \sum_{i=1}^{2}\rho_{i+6,i}&0&0&\sum_{i=7}^{8}\rho_{i,i}
\end{array}\right]
\end{eqnarray}
Obviously it inherits elements from the 3-qubit density matrix \ref{d3}. Different prepa\-rations of the latter result in the same state of the subsystem. When all of the elements of the compound density matrix that participate in the sums of the reduced density matrix are equal, we call the state ``balanced'' (see the first preparation of Bell states in \ref{a3q2}, \ref{a3q3}, \ref{a4q2}, \ref{a4q3}).
Firstly we focus on product states of 3-qubits which result, obviously, in 2-qubit product states. In order to study the entanglement evolution of product states, we calculated the area included between the concurrence graph and the time axis until concurrence reaches zero for the first time after its production, for different values of noise strength. We are interested here in the first cycle of creation-decay of entanglement, which has the largest amount of the latter, and not for any rebirths. This measure is sensitive in all kinds of behavior the entanglement production-decay can exhibit (smooth or oscillatory) and carries information for both of the maximum of entanglement production and the disentanglement time.
The results are presented in the \tref{a3q1}. There are two non-interesting cases where all qubits are aligned ($|eee\rangle$ and $|ggg\rangle$) and noise degrades entanglement monotonically regardless of its placement, two states which present S.AR. behaviour, two states with multiple resonances (S.R. and S.AR.) and two with N.S. behavior. The results show clearly that all of the above effects occur when the noise is applied on the environmental qubit ($M_z=M_3$).
\begin{figure}
\caption{3 interacting qubits. E.S.D. time for several initial states, as a function of noise parameter. (a) We observe the multiple resonances (S.R and S.AR.) for the initial product state $|geg\rangle\to|ge\rangle$, where the S.R is very sharp and occurs with a very small noise strength, while S.AR. is much smoother and occurs with a larger noise strength. (b) N.S. behavior of the balanced Bell state $|\Phi^+\rangle$. (c) Two different preparations of Bell state $|\Psi^+\rangle$ that exhibit S.R. and N.S. correspondingly. Noise is external (affects the traced out qubit) in all of the above cases. ($\langle n\rangle=0$, $\omega_0=4$, $J=0.2$, $\Delta=0.1$, $\gamma=0.01$)}
\label{3q}
\end{figure}
Then we move on to the 2-qubit Bell states ${|\Phi^+\rangle}, {|\Psi^+\rangle}$ and try several initial 3-qubit preparations that result in them.
Almost all of them exhibit noise shield behaviour when noise lies outside our subsystem except from one. Indeed, there is a preparation of $|\Psi^+\rangle$ which exhibits stochastic resonance when noise is environmental (\ref{a3q2}, \ref{a3q3}, \fref{3q} (b,c)).
Finally we present our results for the bipartite subsystem of W state
\begin{eqnarray}
\frac{|eeg\rangle+|ege\rangle+|gee\rangle}{\sqrt{3}},
\end{eqnarray}
which has been proposed as the maximum 3-qubit entangled state and has non-zero entanglement across any bipartition. The W state has matrix representation
\begin{eqnarray}\rho_{W}=\left[ \begin{array}{cccccccc}
0&0 &0 &0 &0&0&0&0\\
0&\frac{1}{3} &\frac{1}{3}&0&\frac{1}{3}&0&0&0 \\
0&\frac{1}{3} &\frac{1}{3} &0&\frac{1}{3}&0&0&0 \\
0&0 &0 &0&0&0&0&0 \\
0&\frac{1}{3} &\frac{1}{3} &0&\frac{1}{3}&0&0&0 \\
0&0 &0 &0&0&0&0&0 \\
0&0 &0 &0&0&0&0& 0\\
0&0 &0 &0&0&0&0&0
\end{array}\right]\end{eqnarray}
and its reduced $2$-qubit density matrix is
\begin{eqnarray}\rho^{r}=\left[ \begin{array}{cccccccc}
\frac{1}{3}&0 &0 &0 \\
0&\frac{1}{3} &\frac{1}{3}&0 \\
0&\frac{1}{3} &\frac{1}{3} &0\\
0&0 &0 &0
\end{array}\right],\end{eqnarray}
with initial entanglement $C_{initial}\simeq0.667$. Again, we observe S.AR. only when the noisy qubit is environmental. The most interesting result, though, is that if we prepare the same $2$-qubit state, but with different elements of the $3$-qubit density matrix, we observe N.S. behavior (\fref{3q2} (a,b)). This can be done easily by setting $\rho_{2,3}=\rho_{3,2}=0$ in the W state preparation. The reduced density matrix (\ref{3q1}) does not contain these elements, so the $2$-qubit state remains the same. The ambiguity in the preparation of the subsystem has important consequences on its evolution.
\begin{figure}
\caption{3 interacting qubits. (a) E.S.D. time for the state resulting after tracing out the third qubit of the W state $\frac{|eeg\rangle+|ege\rangle+|gee\rangle}
\label{3q2}
\end{figure}
\section{4-qubit chain with $T=0$}
The reduced density matrix in this case is:
$$\rho^{r}=\left[ \begin{array}{cccccc}
\sum_{i=1}^4\rho_{i,i}&0&0&\sum_{i=1}^{4}\rho_{i,i+12}\\ 0&\sum_{i=5}^{8}\rho_{i,i}&\sum_{i=5}^{8}\rho_{i,i+4}&0\\0&\sum_{i=5}^{8}\rho_{i+4,i}& \sum_{i=9}^{12}\rho_{i,i}&0\\ \sum_{i=1}^{4}\rho_{i+12,i}&0&0&\sum_{i=13}^{16}\rho_{i,i}
\end{array}\right]$$
The main difference from the 3-qubit case is that the qubits do not interact directly with each other.Qubit $1$ does not interact directly with $3$ (same for $2$ and $4$). We continue to study the entanglement between $1$ and $2$, because we are interested in the behavior of two directly interacting qubits.
As we can see in \tref{a4q1}, there are many product states where the combination of entanglement production and disentanglement time increases monotonically over noise, when the latter is environmental. This is noise shield behavior (\fref{4q1} (a)). We also observe three cases where multiple resonances become evident. Two of them ($|ggeg\rangle, |ggge\rangle$) ``do not obey the rule'' and exhibit resonances only when the noise penetrates at least one qubit of the subsystem under study.
The behavior of Bell states $|\Phi^+\rangle$ and $|\Psi^+\rangle$ is not the same any more. While in the 3-qubit case Bell states $|\Phi^+\rangle$ exhibit N.S. for all of the initial preparations we have studied, here there are some initial preparations that cause the system to exhibit stochastic anti-resonance, similarly to the W state case of 3-qubits (\ref{a4q2}). Furthermore, we found that the N.S. becomes weaker when the anisotropy parameter $\Delta$ increases (\fref{4q1} (b)).
\begin{figure}
\caption{4 interacting qubits (a) N.S. for the product state $|eggg\rangle\to|eg\rangle$. (b) N.S. of the balanced $|\Phi^+\rangle$ against anisotropy constant $\Delta$. The bigger the anisotropy, the weaker the noise shield. ($\langle n\rangle=0$, $\omega_0=4$, $J=0.2$, $\Delta=0.1$, $\gamma=0.01$)}
\label{4q1}
\end{figure}
Bell state $|\Psi\rangle^+$ presents a variety of effects depending on its initial preparation and the placement of noise. We found N.S., S.AR. and S.R. (\ref{a4q3} and \fref{4q12} (a)).
In \fref{4q12} (b) and \fref{4q12} (c) we present the results for the balanced Bell state $|\Psi^+\rangle$. It becomes evident that the application of noise should take place out of the subsystem, in order to create a noise shield. Moreover, the later is amplified when both environmental qubits are noisy, rather than one, something intuitively expected.
\begin{figure}
\caption{4 interacting qubits. (a) E.S.D. time for two preparations of $|\Psi^+\rangle$, when noise affects the two traced out qubits. The black squares correspond to the first preparation (balanced state) in \tref{a4q3}
\label{4q12}
\end{figure}
\section{Non-zero temperature and special initial Hilbert subspaces}
As we presented above, there are some initial preparations both in 3 and 4-qubit chains that cause the subsystem to exhibit S.AR or S.R., while other result in N.S. behavior, which is obviously the best result we can get by adding noise to a quantum system.
The identification of these ``special'' regions of initial Hilbert space is a very difficult task, because of the great number of real parameters needed to describe the density matrix of a 3-qubit or 4-qubit system. The XY interaction Hamiltonian and our Master equation may simplify significantly the system of differential equations we need to solve, but even in this case, the number of real independent parameters for $3$ qubits is $31$ while for $4$ qubits is $127$.
Moreover, the choice of working on the bipartite entanglement does not simplify the problem: The evolution of the reduced 2-qubit density matrix has direct dependance on the non participating elements of the full density matrix, something expected, because one solves firstly the full differential system and then traces out the desired number of qubits.
Needless to say that if we wish to study larger systems, then the problem acquires extremely large complexity and becomes practically unsolvable.
There are two ways to approach this problem:
\begin{enumerate}
\item{{\it The theoretical one}: We have to understand how the initial geometry of the full system provides the necessary background for the beneficial (or not) synchronization of two coexisting timescales (dissipation and decoherence). This could lead to the optimized exploitation of noise addition in order to get the best possible result, which is obviously a noise shield. We believe that the solution of this problem will come from a deeper analysis of the geometry of quantum states and the multipartite entanglement measures.}
\item{{\it The practical one}: Given the complexity of the problem, it would be helpful if we could control the extent of these areas by means of an external macroscopical variable.}
\end{enumerate}
\begin{figure}
\caption{4 interacting qubits (a) E.S.D time for the balanced Bell state $|\Psi^+\rangle$ as a function of temperature parameter $\langle n\rangle$, for different values of $M_{34}
\label{4q3}
\end{figure}
In the present stage of our understanding of the problem of which classes of states are related to noise induced effects, we think that finding an external control seems more straightforward. One option is to study the effect of temperature on the entanglement evolution, since temperature has the advantage of being an easily controllable macroscopical variable.
In our previous work \cite{Ghikas} we observed the negative role of a small increase of temperature in two qubits of XY model (\fref{2q1} (a)). By moving to a larger system ($3$ and $4$ qubits), we expected to see shielding effects, due to the environmental qubits.
This was indeed our first result: The larger the fermionic environment we have, the bigger the tolerance of the subsystem against noise. Although the disentanglement time decreases as $\langle n\rangle$ gets bigger, the results are the same for all of the states which exhibit monotonic behavior in $T=0$ case: Noise shield for $T=0$ remains shield for $T>0$ as we can see in \fref{4q3} (a). We note again that in the case of $4$ qubits the shield becomes ``stronger'' when both of the external qubits are noisy.
The most interesting result though, is the effect of temperature increase on those special initial preparations that result in S.AR. behavior. In the 4-qubit case there is a $\langle n\rangle_{critical}$ under which S.AR. becomes evident. Above this value, the subsystem acquires N.S. behavior as we can see in \fref{4q3} (b). Temperature eliminates the first-negative part of S.AR. and leaves only the second-positive part (ie. N.S.). This is a very welcome result, because it gives us the opportunity to get the best possible noise effect, the noise shield, whether the subsystem begins from a ``S.AR. region'' or not and provides a way to avoid the identification of the latter: If we observe S.AR. then we must increase the temperature and get N.S.!
\section{Conclusions}
In our study of a system of two interacting qubits, under the influence of a bosonic environment and classical noise, we had observed that the appearance of effects of non-monotonic dependence of the disentanglement time on the noise strength, was dependent on the initial preparations of the system \cite{Ghikas}. Then the natural question that arose was whether the initial preparations of the local environment of the 2-qubit subsystem does influence the dynamical evolution. As it has turned out, the local fermionic environment does indeed influence the way the subsystem reacts to an external classical noise.
We have studied several initial preparations of 3-qubit and 4-qubit chains and focused on the bipartite entanglement of the reduced subsystem. Even though they differ by only one qubit, they represent two different cases of a quantum system: In the $3$-qubit chain all qubits interact directly with each other as we see in \fref{a1}. This is not the case in general for Heisenberg type $n$-qubit chains with $n\geq4$. Our system is modelled by a Markovian Master equation with two Lindblad dissipators: one representing the common quantum bath and one representing the classical external field. While the first one is related to the amplitude damping channel, the second one implies a continuous indirect measurement of $V_z$. In \cite{Breuer} one can see that the last term of the Master equation points to the quantum Zeno effect \cite{Misra}, which has been proposed as a reliable method for the protection of entanglement against environmental noise \cite{Zaffino,Francica}.
What we have observed, is that the application of external noise in combination with the proper initial preparation of the compound system, has clear influence on the entanglement evolution of the subsystem.
While many initial preparations of a compound system result in the same subsystem state, they may lead to very different dynamical behaviours of the subsystem entanglement.
The $3$ interesting effects we observed, are:
\begin{itemize}
\item{{\it Noise shield}: Monotonous increase of disentanglement time against noise. This is the most likely effect, according to our summary tables, when noise is environmental (both in $3$ and $4$-qubit chains). The traced out qubit(s) play the role of a local fermionic environment which shields the subsystem against dissipation and decoherence. It is strongly related to quantum Zeno effect.}
\item{{\it Stochastic anti-resonance}: Non-monotonic behavior of the disentanglement time, firstly decreasing and then increasing. This means that a moderate value of noise has the worst possible effect on the system and needs to be avoided. This was initially observed in our 2-qubit study \cite{Ghikas} for various initial states. In our 3 and 4 qubit study, the effects depend not only on the preparation but also on the way the noise is applied. We observed S.AR. for $4$ initial preparations in the 3-qubit chain case, with noise acting only on the traced out qubit. In the 4-qubit case, there were $9$ appearances of S.AR., $7$ with noise acting only on the environmental qubits and $2$ with noise inside the subsystem.}
\item{{\it Stochastic resonance}: Non-monotonic behavior of the disentanglement time, firstly increasing and then decreasing. There is a moderate value of noise that upgrades system reaction. We observed it $3$ times in 3-qubit chains, with noise acting on the traced out qubit and $5$ times in 4-qubit chains, $3$ with noise inside the subsystem and $2$ with noise out of the subsystem.}
\end{itemize}
We note that product and entangled states have different time evolution. Product states begin at zero entanglement, they evolve to entangled states because of qubit interaction and finally loose their entanglement in the presence of environmental decoherence-dissipation. We studied the area under the concurrence graph of the first-largest cycle of entanglement, in order to gain information for both the magnitude of entanglement production and the disentanglement time. Indeed, there is no unique measure of stochastic resonance-like effects and one has to decide which is the best for the physical property under study.
Moreover, temperature can play a constructive role for the control of the special initial Hilbert subspaces which lead to non-monotonic noise effects. It provides us with an easy way to cut out the first-part of S.AR..
Of course the number of different preparations we tried is not large, but it is enough to show the main results, which strengthen our belief that:
\begin{enumerate}
\item{Noise effects are not fully understood yet, even in the Markovian regime which simplifies significantly the mathematical calculations.}
\item{Initial system preparation (and hence the local geometry of the state space) may allow noise to affect decoherence in a positive or negative way.}
\item{The initial local state space of qubit systems needs to be mapped according to the noise effects it can exhibit. The classification of these subspaces could lead to predictions of non-mo\-no\-tonic behavior, and consequently to effective optimization of a quantum computation process.}
\end{enumerate}
\appendix
\section{3-qubit chain density matrix and summary tables}
The general form of the 3-qubit density matrix is:
\begin{eqnarray}\label{d3}
\rho_{s}=\left[ \begin{array}{*{20}{c}}
\bullet &&& \bullet && \bullet & \bullet &\\
& \bullet & \bullet && \bullet &&& \bullet \\
& \bullet & \bullet && \bullet &&& \bullet \\
\bullet &&& \bullet && \bullet & \bullet &\\
& \bullet & \bullet && \bullet &&& \bullet \\
\bullet &&& \bullet && \bullet & \bullet &\\
\bullet &&& \bullet && \bullet & \bullet &\\
& \bullet & \bullet && \bullet &&& \bullet
\end{array} \right],
\end{eqnarray}
where all blank elements are equal to zero.
Next we present summary tables for the 3-qubit case, for several initial preparations of product states and Bell states.
\begin{table}[ht8]
\caption{\label{a3q1} Product state results}
\begin{indented}
\item[]\begin{tabular}{@{} l l l l l}
\br
Initial State & N. S. & S. AR. & S. R. & Noise \\
\mr
$|eee\rangle\to |ee\rangle_{12}$ & & & & \\\mr
$|eeg\rangle\to |ee\rangle_{12}$ & \checkmark & & & $M_3$ \\\mr
$|ege\rangle\to |eg\rangle_{12}$ & &\checkmark & & $M_3$\\\mr
$|egg\rangle\to |eg\rangle_{12}$ & &\checkmark &\checkmark & $M_3$\\\mr
$|gee\rangle\to |ge\rangle_{12}$ & &\checkmark& & $M_3$\\\mr
$|geg\rangle\to |ge\rangle_{12}$ & & \checkmark &\checkmark & $M_3$ \\\mr
$|gge\rangle\to |gg\rangle_{12}$ & \checkmark & & & $M_3$\\\mr
$|ggg\rangle\to |gg\rangle_{12}$ & & & & \\
\br
\end{tabular}
\end{indented}
\end{table}
\begin{table}[ht7]
\caption{\label{a3q2} Bell state $|\Phi^+\rangle=\frac{|ee\rangle+|gg\rangle}{\sqrt{2}}$}
\begin{indented}
\item[]\begin{tabular}{@{} l l l l l}
\br
Initial State & N. S. & S. AR. & S. R. & Noise \\
\mr
$\rho_{1,1}=\rho_{2,2}=\rho_{7,7}=\rho_{8,8}=1/4$& \checkmark & & & $M_3$ \\
$\rho_{1,7}=\rho_{7,1}=\rho_{2,8}=\rho_{8,2}=1/4$&&&&\\\mr
$\rho_{1,1}=0,\rho_{2,2}=1/2,\rho_{7,7}=0,\rho_{8,8}=1/2$& \checkmark & & & $M_3$ \\
$\rho_{1,7}=\rho_{7,1}=0,\rho_{2,8}=\rho_{8,2}=1/2$&&&&\\\mr
$\rho_{1,1}=2/5,\rho_{2,2}=1/10,\rho_{7,7}=2/5,\rho_{8,8}=1/10$& \checkmark & & & $M_3$ \\
$\rho_{1,7}=\rho_{7,1}= 2/5,\rho_{2,8}=\rho_{8,2}=1/10$&&&&\\\mr
$\rho_{1,1}=1/10,\rho_{2,2}=2/5,\rho_{7,7}=1/10,\rho_{8,8}=2/5$& \checkmark & & & $M_3$ \\
$\rho_{1,7}=\rho_{7,1}= 1/10,\rho_{2,8}=\rho_{8,2}=2/5$&&&&\\\mr
$\rho_{1,1}=1/2,\rho_{2,2}=0,\rho_{7,7}=1/2,\rho_{8,8}=0$& \checkmark & & & $M_3$ \\
$\rho_{1,7}=\rho_{7,1}= 1/2,\rho_{2,8}=\rho_{8,2}=0$&&&&\\
\br
\end{tabular}
\end{indented}
\caption{\label{a3q3} Bell state $|\Psi^+\rangle=\frac{|eg\rangle+|ge\rangle}{\sqrt{2}}$}
\begin{indented}
\item[]\begin{tabular}{@{} l l l l l}
\br
Initial State & N. S. & S. AR. & S. R. & Noise \\
\mr
$\rho_{3,3}=\rho_{4,4}=\rho_{5,5}=\rho_{6,6}=1/4$& \checkmark & & & $M_3$ \\
$\rho_{3,5}=\rho_{5,3}=\rho_{4,6}=\rho_{6,4}=1/4$&&&&\\\mr
$\rho_{3,3}=0,\rho_{4,4}=1/2,\rho_{5,5}=0,\rho_{6,6}=1/2$& & & \checkmark & $M_3$ \\
$\rho_{3,5}=\rho_{5,3}=0,\rho_{4,6}=\rho_{6,4}=1/2$&&&&\\\mr
$\rho_{3,3}=1/2,\rho_{4,4}=0,\rho_{5,5}=1/2,\rho_{6,6}=0$& \checkmark & & & $M_3$ \\
$\rho_{3,5}=\rho_{5,3}=1/2,\rho_{4,6}=\rho_{6,4}=0$&&&&\\\mr
$\rho_{3,3}=2/5,\rho_{4,4}=1/10,\rho_{5,5}=2/5,\rho_{6,6}=1/10$& \checkmark & & & $M_3$ \\
$\rho_{3,5}=\rho_{5,3}=2/5,\rho_{4,6}=\rho_{6,4}=1/10$&&&&\\\mr
$\rho_{3,3}=1/10,\rho_{4,4}=2/5,\rho_{5,5}=1/10,\rho_{6,6}=2/5$& \checkmark & & & $M_3$ \\
$\rho_{3,5}=\rho_{5,3}=1/10,\rho_{4,6}=\rho_{6,4}=2/5$&&&&\\
\br
\end{tabular}
\end{indented}
\end{table}
\section{4-qubit chain density matrix and summary tables}
The general form of the 4-qubit density matrix is:
\begin{eqnarray}\label{d4}
\rho_{s}=
\left[ \begin{array}{*{20}{c}}
\bullet &&& \bullet && \bullet & \bullet &&& \bullet & \bullet && \bullet &&& \bullet \\
& \bullet & \bullet && \bullet &&& \bullet & \bullet &&& \bullet && \bullet & \bullet &\\
& \bullet & \bullet && \bullet &&& \bullet & \bullet &&& \bullet && \bullet & \bullet &\\
\bullet &&& \bullet && \bullet & \bullet &&& \bullet & \bullet && \bullet &&& \bullet \\
& \bullet & \bullet && \bullet &&& \bullet & \bullet &&& \bullet && \bullet & \bullet &\\
\bullet &&& \bullet && \bullet & \bullet &&& \bullet & \bullet && \bullet &&& \bullet \\
\bullet &&& \bullet && \bullet & \bullet &&& \bullet & \bullet && \bullet &&& \bullet \\
& \bullet & \bullet && \bullet &&& \bullet & \bullet &&& \bullet && \bullet & \bullet &\\
& \bullet & \bullet && \bullet &&& \bullet & \bullet &&& \bullet && \bullet & \bullet &\\
\bullet &&& \bullet && \bullet & \bullet &&& \bullet & \bullet && \bullet &&& \bullet \\
\bullet &&& \bullet && \bullet & \bullet &&& \bullet & \bullet && \bullet &&& \bullet \\
& \bullet & \bullet && \bullet &&& \bullet & \bullet &&& \bullet && \bullet & \bullet &\\
\bullet &&& \bullet && \bullet & \bullet &&& \bullet & \bullet && \bullet &&& \bullet \\
& \bullet & \bullet && \bullet &&& \bullet & \bullet &&& \bullet && \bullet & \bullet &\\
& \bullet & \bullet && \bullet &&& \bullet & \bullet &&& \bullet && \bullet & \bullet &\\
\bullet &&& \bullet && \bullet & \bullet &&& \bullet & \bullet && \bullet &&& \bullet
\end{array} \right],
\end{eqnarray}
where all blank elements are equal to zero.
Next we present summary tables for the 4-qubit case, for several initial preparations of product states and Bell states.
\begin{table}[ht4]
\caption{\label{a4q1} Product states}
\begin{indented}
\item\begin{tabular}{@{} l l l l l p{6cm}}
\br
Initial State & N. S. & S. AR. & S. R. & Noise \\
\mr
$|eeee\rangle\to |ee\rangle_{12}$ & & & \\
\mr
$|eeeg\rangle\to |ee\rangle_{12}$ & \checkmark & & &$M_{34},M_4$ \\
\mr
$|eege\rangle\to |ee\rangle_{12}$ & \checkmark & & & $M_{34}$ \\
\mr
$|eegg\rangle\to |ee\rangle_{12}$ & \checkmark & & & $M_{34},M_4$ \\
\mr
$|egee\rangle\to |ee\rangle_{12}$ & \checkmark & & & $M_{34},M_4$ \\
\mr
$|egeg\rangle\to |eg\rangle_{12}$ & &\checkmark & & $M_{34},M_4$ \\
\mr
$|egge\rangle\to |eg\rangle_{12}$ & \checkmark & & & $M_{34},M_4$ \\
\mr
$|eggg\rangle\to |eg\rangle_{12}$ & \checkmark & & & $M_{34},M_4$ \\
\mr
$|geee\rangle\to |ge\rangle_{12}$ & \checkmark & & & $M_{34},M_4$ \\
\mr
$|geeg\rangle\to |ge\rangle_{12}$ & \checkmark & & & $M_{34},M_4$ \\
\mr
$|gege\rangle\to |ge\rangle_{12}$ & \checkmark &\checkmark & & N.S.$\to M_4$, S.AR. $\to M_{34}$ \\
\mr
$|gegg\rangle\to |ge\rangle_{12}$ & \checkmark & \checkmark & \checkmark & N.S $\to$ $M_{34}$, S.AR. + S.R. $\to$ $M_{4}$ \\
\mr
$|ggee\rangle\to |gg\rangle_{12}$ & \checkmark & & & $M_{34}$, $M_{4}$ \\
\mr
$|ggeg\rangle\to |gg\rangle_{12}$ & &\checkmark & \checkmark & S.AR. + S.R. $\to$ $M_{1234}$, \\
& & & & $M_{234}$, $M_{12}$, $M_{13}$,\\& & & & $M_{14}$, $M_1$\\
\mr
$|ggge\rangle\to |gg\rangle_{12}$ &\checkmark & \checkmark & \checkmark & N.S $\to$ $M_{4}$\\
& & & & S.AR. + S.R. $\to$ $M_{1234}$, \\
& & & & $M_{234}$, or $M_{12}$, $M_{13}$,\\& & & & $M_{14}$, $M_{1}$\\
\mr
$|gggg\rangle\to |gg\rangle_{12}$ & & & & \\
\br
\end{tabular}
\end{indented}
\end{table}
S.AR. + S.R. means that there exist resonance and anti-resonance in the same diagram (i.e. \fref{3q}(a)).
\begin{table}[ht5]
\caption{\label{a4q2} Bell state $|\Phi^+\rangle=\frac{|ee\rangle+|gg\rangle}{\sqrt{2}}$}
\begin{indented}
\item[]\begin{tabular}{@{} l l l l l}
\br
Initial State & N. S. & S. AR. & S. R. & Noise \\
\mr
$\rho_{1,1}=\rho_{2,2}=\rho_{3,3}=\rho_{4,4}=\frac{1}{8}$& &&& \\
$\rho_{13,13}=\rho_{14,14}=\rho_{15,15}=\rho_{16,16}=\frac{1}{8}$&\checkmark & & &$M_{34}$, $M_4$\\
$\rho_{13,1}=\rho_{14,2}=\rho_{15,3}=\rho_{16,4}=\frac{1}{8}$&&&&\\
$\rho_{1,13}=\rho_{2,14}=\rho_{3,15}=\rho_{4,16}=\frac{1}{8}$&&&&\\\mr
$\rho_{1,1}=\frac{1}{2},\rho_{2,2}=\rho_{3,3}=\rho_{4,4}=0$& &&& \\
$\rho_{13,13}=\frac{1}{2},\rho_{14,14}=\rho_{15,15}=\rho_{16,16}=0$& &\checkmark & &$M_{34}$ \\
$\rho_{13,1}=\rho_{1,13}=\frac{1}{2},\rho_{14,2}=\rho_{15,3}=0$&&&&\\
$\rho_{2,14}=\rho_{3,15}=\rho_{4,16}=\rho_{16,4}=0$&&&&\\\mr
$\rho_{1,1}=0,\rho_{2,2}=\frac{1}{2},\rho_{3,3}=\rho_{4,4}=0$& &&& \\
$\rho_{14,14}=\frac{1}{2},\rho_{13,13}=\rho_{15,15}=\rho_{16,16}=0$&\checkmark & & &$M_{34}$, $M_4$ \\
$\rho_{13,1}=\rho_{1,13}=0,\rho_{2,14}=\rho_{14,2}=\frac{1}{2},$&&&&\\
$\rho_{3,15}=\rho_{15,3}=\rho_{4,16}=\rho_{16,4}=0$&&&&\\\mr
$\rho_{1,1}=\rho_{2,2}=0,\rho_{3,3}=\rho_{4,4}=\frac{1}{4}$& &&& \\
$\rho_{13,13}=\rho_{14,14}=0,\rho_{15,15}=\rho_{16,16}=\frac{1}{4}$&\checkmark & & &$M_{34}$, $M_4$ \\
$\rho_{13,1}=\rho_{1,13}=\rho_{2,14}=\rho_{14,2}=0,$&&&&\\
$\rho_{3,15}=\rho_{15,3}=\rho_{4,16}=\rho_{16,4}=\frac{1}{4}$&&&&\\\mr
$\rho_{1,1}=\rho_{2,2}=\frac{1}{4},\rho_{33}=\rho_{4,4}=0$& &&& \\
$\rho_{13,13}=\rho_{14,14}=\frac{1}{4},\rho_{15,15}=\rho_{16,16}=0$&\checkmark & & &$M_{34}$, $M_4$ \\
$\rho_{13,1}=\rho_{1,13}=\rho_{2,14}=\rho_{14,2}=\frac{1}{4}$&&&&\\
$\rho_{3,15}=\rho_{15,3}=\rho_{4,16}=\rho_{16,4}=0$&&&&\\\mr
$\rho_{1,1}=\rho_{2,2}=\rho_{3,3}=0,\rho_{4,4}=\frac{1}{2}$& &&& \\
$\rho_{13,13}=\rho_{14,14}=\rho_{15,15}=0,\rho_{16,16}=\frac{1}{2}$& &\checkmark & & $ M_{34}$ \\
$\rho_{13,1}=\rho_{1,13}=\rho_{2,14}=\rho_{14,2}=0$&&&&\\
$\rho_{3,15}=\rho_{15,3}=0,\rho_{4,16}=\rho_{16,4}=\frac{1}{2}$&&&&\\\mr
$\rho_{1,1}=\rho_{2,2}=\frac{1}{16},\rho_{3,3}=\rho_{4,4}=\frac{3}{16}$& &&& \\
$\rho_{13,13}=\rho_{14,14}=\frac{1}{16},\rho_{15,15}=\rho_{16,16}=\frac{3}{16}$&\checkmark & & & $ M_{34},M_4$ \\
$\rho_{13,1}=\rho_{1,13}=\rho_{2,14}=\rho_{14,2}=\frac{1}{16}$&&&&\\
$\rho_{3,15}=\rho_{15,3}=\rho_{4,16}=\rho_{16,4}=\frac{3}{16}$&&&&\\
\br
\end{tabular}
\end{indented}
\end{table}
\begin{table}[ht]
\caption{ \label{a4q3}Bell state $|\Psi^+\rangle=\frac{|eg\rangle+|ge\rangle}{\sqrt{2}}$}
\begin{tabular}{@{} l l l l l}
\br
Initial State & N.S. & S.AR. & S.R. & Noise \\
\mr
$\rho_{5,5}=\rho_{6,6}=\rho_{7,7}=\rho_{8,8}=\frac{1}{8}$& &&& \\
$\rho_{9,9}=\rho_{10,10}=\rho_{11,11}=\rho_{12,12}=\frac{1}{8}$&\checkmark & & &$M_{34},M_4$\\
$\rho_{5,9}=\rho_{6,10}=\rho_{7,11}=\rho_{8,12}=\frac{1}{8}$&&&&\\
$\rho_{9,5}=\rho_{10,6}=\rho_{11,7}=\rho_{12,8}=\frac{1}{8}$&&&&\\\mr
$\rho_{5,5}=\rho_{6,6}=\frac{3}{16},\rho_{7,7}=\rho_{8,8}=\frac{1}{16}$& &&& \\
$\rho_{9,9}=\rho_{10,10}=\frac{3}{16},\rho_{11,11}=\rho_{12,12}=\frac{1}{16}$&\checkmark & & &$M_{34},M_4$\\
$\rho_{5,9}=\rho_{6,10}=\frac{3}{16},\rho_{7,11}=\rho_{8,12}=\frac{1}{16}$&&&&\\
$\rho_{9,5}=\rho_{10,6}=\frac{3}{16},\rho_{11,7}=\rho_{12,8}=\frac{1}{16}$&&&&\\\mr
$\rho_{5,5}=\rho_{6,6}=\frac{1}{16},\rho_{7,7}=\rho_{8,8}=\frac{3}{16}$& &&& \\
$\rho_{9,9}=\rho_{10,10}=\frac{1}{16},\rho_{11,11}=\rho_{12,12}=\frac{3}{16}$&\checkmark & & &$M_{34},M_4$\\
$\rho_{5,9}=\rho_{6,10}=\frac{1}{16},\rho_{7,11}=\rho_{8,12}=\frac{3}{16}$&&&&\\
$\rho_{9,5}=\rho_{10,6}=\frac{1}{16},\rho_{11,7}=\rho_{12,8}=\frac{3}{16}$&&&&\\\mr
$\rho_{5,5}=\frac{1}{2},\rho_{6,6}=\rho_{7,7}=\rho_{8,8}=0$& &&& \\
$\rho_{9,9}=\frac{1}{2},\rho_{10,10}=\rho_{11,11}=\rho_{12,12}=0$&\checkmark & & &$M_{34},M_4$\\
$\rho_{5,9}=\frac{1}{2},\rho_{6,10}=\rho_{7,11}=\rho_{8,12}=0$&&&&\\
$\rho_{9,5}=\frac{1}{2},\rho_{10,6}=\rho_{11,7}=\rho_{12,8}=0$&&&&\\\mr
$\rho_{5,5}=0,\rho_{6,6}=\frac{1}{2},\rho_{7,7}=\rho_{8,8}=0$& &&& \\
$\rho_{9,9}=0,\rho_{10,10}=\frac{1}{2},\rho_{11,11}=\rho_{12,12}=0$& &\checkmark & &$M_{34},M_4$ \\
$\rho_{5,9}=0,\rho_{6,10}=\frac{1}{2},\rho_{7,11}=\rho_{8,12}=0$&&&&\\
$\rho_{9,5}=0,\rho_{10,6}=\frac{1}{2},\rho_{11,7}=\rho_{12,8}=0$&&&&\\\mr
$\rho_{5,5}=\rho_{6,6}=0,\rho_{7,7}=\frac{1}{2},\rho_{8,8}=0$& &&& \\
$\rho_{9,9}=\rho_{10,10}=0,\rho_{11,11}=\frac{1}{2},\rho_{12,12}=0$& &\checkmark & \checkmark &S.AR. $\to M_{34}$ \\&&&&S.R.+S.AR. $\to M_4$ \\
$\rho_{5,9}=\rho_{6,10}=0,\rho_{7,11}=\frac{1}{2},\rho_{8,12}=0$&&&&\\
$\rho_{9,5}=\rho_{10,6}=0,\rho_{11,7}=\frac{1}{2},\rho_{12,8}=0$&&&&\\\mr
$\rho_{5,5}=\rho_{6,6}=\rho_{7,7}=0,\rho_{8,8}=\frac{1}{2}$ \\
$\rho_{9,9}=\rho_{10,10}=\rho_{11,11}=0,\rho_{12,12}=\frac{1}{2}$&\checkmark & & \checkmark &\parbox[t]{4cm}{N.S $\to M_{34},M_4$\\ S.R. $\to M_{234},M_2$} \\
$\rho_{5,9}=\rho_{6,10}=\rho_{7,11}=0,\rho_{8,12}=\frac{1}{2}$\\
$\rho_{9,5}=\rho_{10,6}=\rho_{11,7}=0,\rho_{12,8}=\frac{1}{2}$\\
\br
\end{tabular}
\end{table}
\section*{References}
\end{document} | math |
ટોળાએ શાળામાં ઘુસી શિક્ષિકાને માર્યો માર, કપડાં પણ ફાડી નાખ્યા: ભાજપે હિજાબનો એંગલ જણાવ્યો પશ્ચિમ બંગાળના દક્ષિણ દિનાજપુર જિલ્લામાં, તેના પરિવારના સભ્યોએ શાળાની અંદર ઘૂસીને એક મહિલા શિક્ષિકાને ઠપકો આપવા માટે હુમલો કર્યો. એટલું જ નહીં આ દરમિયાન તેણે પોતાના કપડા ફાડી નાખ્યા. શુક્રવારે બપોરે બાળકીના પરિવાર સાથે આવેલા ટોળાએ શિક્ષકોના કોમન રૂમમાં ઘૂસીને મહિલા શિક્ષક સાથે ગેરવર્તન કર્યું હતું. આ દરમિયાન તેઓએ શિક્ષકને માર માર્યો હતો અને તેના કપડાં પણ ફાડી નાખ્યા હતા. આ મામલો શુક્રવારે બપોરે જણાવવામાં આવી રહ્યો છે. હિલી પોલીસ સ્ટેશન હેઠળ આવતી ત્રિમોહિની પ્રતાપ ચંદ્ર હાઈસ્કૂલના શિક્ષકે ગુરુવારે ક્લાસમાં ન આવવા પર વિદ્યાર્થિનીને ઠપકો આપ્યો હતો અને તેને થપ્પડ પણ મારી હતી. છોકરીએ ઘરે જઈને પરિવારના સભ્યોને ફરિયાદ કરી, જેમણે બીજા દિવસે ટોળા સાથે શાળામાં ધસી આવ્યા. શુક્રવારે બપોરે એક ટોળું શાળામાં ઘુસી ગયું હતું અને શિક્ષકોના કોમન રૂમમાં જઈને તેમને માર માર્યો હતો. આ દરમિયાન ટોળાએ મહિલા શિક્ષિકાને પણ અપશબ્દો બોલ્યા હતા. ઘટનાની જાણ થતા પોલીસ કાફલો પણ ઘટના સ્થળે પહોંચી ગયો હતો. શાળા પ્રશાસન વતી ફરિયાદ નોંધ્યા બાદ પોલીસે રવિવારે આ મહિલા શિક્ષિકા સાથે ગેરવર્તન કરવા બદલ 4 લોકોની ધરપકડ કરી હતી. બંગાળમાં પણ આ મુદ્દાએ રાજકીય વળાંક લીધો છે. રવિવારે બીજેપી પ્રદેશ અધ્યક્ષ સુકાંત મજમુદાર પણ ઘટનાસ્થળે પહોંચ્યા હતા અને વિરોધ કરી રહેલા લોકો સાથે મુલાકાત કરી હતી. પોલીસે આવા કૃત્યો કરનારાઓ સામે કડક કાર્યવાહી કરવી જોઈએ તેવી માંગણી કરી હતી. બીજેપી અધ્યક્ષે કહ્યું વિદ્યાર્થીનો હિજાબ ઉતારતા ટોળાએ હુમલો કર્યો આ ઘટનાનો વીડિયો ભાજપ યુવા મોરચાના પ્રદેશ ઉપાધ્યક્ષ તરુણજ્યોતિ તિવારીએ ટ્વિટર પર શેર કર્યો છે. આ ઘટના બાદ સ્થાનિક લોકો નારાજ છે અને રવિવારે પણ તેઓએ વિરોધ પ્રદર્શન કર્યું હતું. ઘટનાસ્થળે પહોંચેલા બીજેપી પ્રદેશ અધ્યક્ષે પણ આ ઘટનાને હિજાબ સાથે જોડી દીધી. તેણે આરોપ લગાવ્યો, હું પણ એક સમયે શિક્ષક હતો. અમે વિદ્યાર્થીઓને શિસ્ત પણ આપતા. શિક્ષક માટે, તેનો વિદ્યાર્થી માત્ર વિદ્યાર્થી છે, બીજું કંઈ નથી. શિક્ષકે છોકરીનો કાન પકડી લીધો હતો અને આ દરમિયાન તેનું હિજાબ નીચે પડી ગયું હતું. જેના કારણે આ ઘટના બની છે. શિક્ષકે કહ્યું, અમે અસુરક્ષિત અનુભવીએ છીએ, 200ના ટોળાએ હુમલો કર્યો તેમણે કહ્યું કે આટલી જ વાત પર છોકરીના પરિવારજનોએ 200 લોકોની ભીડ સાથે શાળામાં ધસી આવ્યા હતા. પોલીસે આ મામલે ગુનો પણ નોંધ્યો ન હતો. જ્યારે સ્થાનિક લોકો વતી હોબાળો થયો, ત્યારે કેસ નોંધવામાં આવ્યો. પોલીસે 35 લોકો વિરુદ્ધ FIR નોંધી છે. હુમલાનો ભોગ બનેલા શિક્ષકે કહ્યું, જ્યારે વિદ્યાર્થીઓ વર્ગમાં અનુશાસનહીનતા બતાવે છે અથવા અભ્યાસમાં ધ્યાન આપતા નથી, તો અમે તેમના કાન પકડી લઈએ છીએ અથવા ક્યારેક તેમને થપ્પડ મારીએ છીએ. આ શિસ્ત માટે કરવામાં આવે છે. હવે આ ઘટના બાદ અમે અસુરક્ષિત અનુભવી રહ્યા છીએ. છેવટે, અમે શાળાના 2,000 વિદ્યાર્થીઓને કેવી રીતે શિસ્ત આપીશું? દ્રૌપદી મુર્મુ બન્યા દેશના 15મા રાષ્ટ્રપતિ, CJI રમણાએ અપાવી શપથ ભારતમાં નવા કોવિડ19 કેસમાં 16.8 ટકાનો ઘટાડો, છેલ્લા 24 કલાકમાં 16,866 કેસ ભૂસ્ખલનને કારણે રામબનમાં અમરનાથ યાત્રા રોકવામાં આવી,જાણો mantavyanews.com Copyright 2021 Mantavya News | gujurati |
నదిజలాల వాటాల్లో తెలంగాణకు తీవ్ర అన్యాయం ప్రత్యేక తెలంగాణ రాష్ట్రం సాధించుకోవడానికి ఉ మ్మడి ఆంధ్రప్రదేశ్ రాష్ట్రంలో 10 జిల్లాల తెలంగాణ ప్రాంతానికి నీళ్ల విషయంలో తీవ్ర అన్యాయం జరుగుతుందని భావించి తెలంగాణ రాష్ట్రం సాధించుకోవడానికి నీళ్ల సమస్య ఒక ప్రధాన భూమిక పోషించిన విషయం తెలిసిందే,ప్రస్తుతం ఆంధ్రప్రదేశ్ మరియు తెలంగాణ రెండు రాష్ట్రాలుగా విడిపోయి పాలన స్వతంత్రంగా సాగుతున్న కానీ ఇప్పటికీ కూడా తెలంగాణ రాష్ట్రానికి రావలసిన వాటాను ఉపయోగించుకోవడం లేదు ఇప్పటికీ కూడా కృష్ణనది నీళ్ల విషయంలో తెలంగాణ ప్రాంతానికి తీవ్ర అన్యాయం జరుగుతుంది దానికి ప్రధాన కారణం తెలంగాణ ముఖ్యమంత్రి కేసీఆర్ ,తెలంగాణ ప్రభుత్వ వైఫల్యం అన్ని ఖచ్చితంగా చెప్పవచ్చు, తెలంగాణ ముఖ్యమంత్రి కేసీఆర్ తీసుకునే తెలంగాణ వ్యతిరేక నిర్ణయాలు అనేది కాంట్రాక్టర్ల కోసమో,లేక కమీషన్ల కోసమో కకృతి పడి తెలంగాణ సమాజాని ఆంధ్రప్రదేశ్ ప్రభుత్వంతో తాకట్టు పెట్టడంతో తెలంగాణ భవిష్యత్ ప్రమాదంలో పడే అవకాశం ఉంది. ఉమ్మడి ఆంధ్రప్రదేశ్ రాష్ట్రంలో కృష్ణానది నీటిని దోచుక పోవడానికి నాటి అంధ్రప్రదేశ్ ముఖ్యమంత్రి డా. వైఎస్ రాజశేఖర్ రెడ్డి పోతిరెడ్డిపాడును తవ్వి తెలంగాణ ప్రాంతానికి నీళ్ల సమస్యను ఏ విధంగా సృష్టించారో ఇప్పుడు అతని కుమారుడు ప్రస్తుత ఆంధ్రప్రదేశ్ ముఖ్యమంత్రి వైయస్ జగన్ కూడా తెలంగాణ ప్రాంతానికి అన్యాయం చేస్తూ ఆంధ్రప్రదేశ్ ప్రాంతంలో అనుమతులు లేకుండా కూడా సంగమేశ్వర,రాయలసీమ ఎత్తిపోతల పథకాలను నిర్మించడం అంటే అది ముమ్మాటికి తెలంగాణ ప్రాంతానికి కృష్ణనది నీళ్ల విషయంలో మరొకసారి ఆంధ్రప్రదేశ్ ప్రభుత్వం ద్రోహం చేస్తున్నట్లే అని స్పష్టంగా అర్థం చేసుకోవచ్చు,వైయస్ రాజశేఖర్ రెడ్డి పోతిరెడ్డిపాడుతో అన్యాయం చేస్తే ఇప్పుడు అతని కుమారుడు వైయస్ జగన్మోహాన్ రెడ్డి సంగమేశ్వర దేవాలయం నుంచి కృష్ణ నీటిని తరలించి మూలిగే నక్కపై తాటిపండు పడ్డట్లు తెలంగాణకు మరింత అన్యాయం చేసేందుకు ఆంధ్రప్రదేశ్ ప్రభుత్వం రెడీ అయింది. ఇంత అన్యాయం జరుగుతున్నా టీఆర్ఎస్ ప్రభుత్వం ఈ కృష్ణ నది నీళ్ల విషయంలో శాశ్వత పరిష్కార మార్గాలు ఎంచుకోకపోవడం తెలంగాణ రాష్ట్రానికి,మరి ముఖ్యంగా ఉమ్మడి మహబూబ్ నగర్,దక్షిణ తెలంగాణ ప్రాంతానికి టీఆర్ఎస్ ప్రభుత్వం అన్యాయం చేస్తుందని చెప్పక తప్పదు, రెండు రాష్ట్రాల మధ్య నీళ్ల విషయంలో పరిష్కారం దిశగా అడుగులు వేసేందుకు కృష్ణ నది యాజమాన్య బోర్డు అనేక సందర్భాలలో సమావేశాలు ఏర్పాటు చేసిన కూడా ఆ సమావేశాలకు సరిగ్గా హాజరు కాకపోవడంతో తెలంగాణకు నది జలాల విషయంలో ఎంత మేరకు అన్యాయం జరుగుతుంది అని వివరించి లేకపోతున్నారు ఇదే అనువుగా చేసుకున్న ఆంధ్రప్రదేశ్ ప్రభుత్వం తెలంగాణ ప్రభుత్వ వ్యవహార శైలి కారణంగా కృష్ణ నది నీటిని దోచుకోవడమే లక్ష్యంగా ఆంధ్రప్రదేశ్ ప్రభుత్వం పని చేస్తుంది,ఇంకా చెప్పాలంటే రాష్ట్రాల మధ్య ఏమైనా సమస్యలు,ఇబ్బందులు ఉంటే వాటిని చెప్పుకుని పరిష్కారం దిశగా అడుగులు వేసేటువంటి అఫ్ఫెక్స్ కౌన్సిల్ సమావేశానికి హాజరు కాకపోవడం వల్లన తెలంగాణ ప్రాంతానికి ఏ మేరకు ఇబ్బంది తలెత్తుతుంది ,దానిని ఏ విధంగా పరిష్కరించాలి అనే ఆలోచన తెలంగాణ ప్రభుత్వానికి లేనప్పుడు కృష్ణ యాజమాన్య బోర్డ్ ఈ సమస్యను ఎలా పరిష్కరిస్తుంది! అపెక్స్ కౌన్సిల్ సమావేశానికి హాజరైతే ఆంధ్రప్రదేశ్ ప్రభుత్వం నిర్మించే అక్రమ ప్రాజెక్టులపై ఫిర్యాదు చేసి వాటిని ఆపే అధికారం,ప్రశ్నించే అధికారం ఈ అపెక్స్ కౌన్సిల్ సమావేశంలో ఉంటుంది అస్సలు తెలంగాణ ప్రభుత్వం ఈ అపెక్స్ కౌన్సిల్ సమావేశానికి హాజరు కానప్పుడు తెలంగాణ రాష్ట్రానికి నీళ్ల వాటా విషయంలో ఏ విధంగా న్యాయం జరుగుతుంది. ఆంధ్రప్రదేశ్ లో ఉన్నటువంటి అనేక ప్రాజెక్టులకు కృష్ణ నది యాజమాన్య బోర్డ్ అనుమతి అనేది రాయలసీమ ప్రాంతానికి తాగునీటి తో పాటు,సాగు నీటికి అనుమతి ఇచ్చిందే వేల టీఎమ్ సీ ల నీళ్లు సముద్రపాలు కాకుండా,వృధా చేయకుండా ఆ నీటిని పరిరక్షించుకోవడమే లక్ష్యంగా ఆ ప్రాజెక్టులకు కృష్ణ యాజమాన్య బోర్డు,కేంద్ర ప్రభుత్వ అనుమతులు ఇచ్చింది, అలాంటప్పుడు వరద వచ్చే సమయంలో వేల టీఎమ్ సీల నీళ్లు,లక్షల క్యూసెక్కుల నీళ్లు సముద్రం పాలు కాకుండా వాటిని కాపాడుకోవడం కోసం మాత్రమే, కానీ ఆంధ్రప్రదేశ్ ప్రభుత్వం ఆ విషయాలన్నిటినీ మర్చిపోయి మరోసారి దక్షిణ తెలంగాణ ప్రాంతానికి ఘోర అన్యాయం చేస్తుంది. దీంతో తెలంగాణ రాష్ట్రానికి రావలసిన వాటా కూడా అంధ్రప్రదేశ్ రాష్ట్రం ఉపయోగించుకుంటుంది ఇది అన్యాయం,అక్రమం కూడా. ఆంధ్రప్రదేశ్ ముఖ్యమంత్రి వై యస్ జగన్మోహన్ రెడ్డి తీసుకునే నిర్ణయాలు,తెలంగాణ ప్రభుత్వ వైఫల్యల వల్ల దక్షిణ తెలంగాణ ప్రాంతం పూర్తిగా ఎడారిగా మారే ప్రమాదం ఉన్నది ఈ ప్రమాదం నుంచి త్వరగా బయట పడాలంటే ఖచ్చితంగా కృష్ణ జలాలను కాపాడే విధంగా,కృష్ణ జలాలను పరిరక్షించే విధంగా తెలంగాణ ప్రభుత్వం తక్షణమే చొరవ తీసుకోవాలి అప్పుడే తెలంగాణ ప్రాంతానికి కృష్ణ నది విషయంలో న్యాయం జరుగుతుంది లేకపోతే ఈ ప్రభుత్వం చేసే వైఫల్యాల కారణంగా తరతరాలుగా ఈ దక్షిణ తెలంగాణ ప్రాంతం ఎడారిగా మారే ప్రమాదం ఉంది.ఈ పరిణామాలన్నింటినీ చాలా నిశితంగా పరిశీలిస్తున్న బిజెపి కేంద్ర ప్రభుత్వం తెలంగాణ ప్రభుత్వం,కేసీఆర్ చేతకానితనాన్ని చూసి రెండు రాష్ట్రాల మధ్య ఉన్నటువంటి కృష్ణానది నీళ్ల సమస్యను పరిష్కారించేందుకు సిద్ధం అయింది. కృష్ణ నది నీళ్ల వాటా విషయంలో 555 ఎల నీళ్లు తెలంగాణ ప్రాంతానికి రావాలి కానీ కేసీఆర్ మాత్రం కేవలం 299 టీఎమ్ సీల కు మాత్రమే అంగీకరించి తెలంగాణ రాష్ట్రానికి 256 టీఎమ్ సీల నీళ్ళను తెలంగాణ రాష్ట్రానికి ద్రోహం చేశారు ,అంతేకాదు ఆంధ్రప్రదేశ్ పునర్విభజన చట్టం 2014 ప్రకారం రెండు రాష్ట్రాలకు కృష్ణా యాజమాన్య బోర్డు ప్రకటించిన నీటి వాటా కంటే దాదాపు 140 నుంచి 150 టీఎమ్ సీల వరకు ఆంధ్రప్రదేశ్ అదనంగా వాడుకుంటుంది,ఇంత అన్యాయం,దారుణం జరుగుతున్న కానీ ముఖ్యమంత్రి కేసీఆర్ ,తెలంగాణ ప్రభుత్వం ఏ మాత్రం చర్య తీసుకోక పోవడం చాలా దురదృష్టకరం. గిరి వర్ధన్ రెడ్డి, బిజెపి మేడ్చల్ జిల్లా ప్రధాన కార్యదర్శి | telegu |
നിപ്പ, കോവിഡ് പ്രതിരോധം: മുക്കത്ത് അവലോകന യോഗം ചേര്ന്നു മുക്കം: ചാത്തമംഗലം മുന്നൂരില് നിപ്പ വൈറസ് സ്ഥിരീകരിച്ച സാഹചര്യത്തിലും തിരുവമ്ബാടി മണ്ഡലത്തിലെ കോവിഡ് പ്രതിരോധ പ്രവര്ത്തനങ്ങള് വിലയിരുത്തുന്നതിനുമായി ലിന്റോ ജോസഫ് എംഎല്എയുടെ അധ്യക്ഷതയില് യോഗം ചേര്ന്നു. മുക്കം നഗരസഭ ചെയര്മാന്, കൊടിയത്തൂര്, കാരശേരി പഞ്ചായത്ത് പ്രസിഡന്റുമാര്, പോലീസ് ഉദ്യോഗസ്ഥര് എന്നിവര് സംബധിച്ച യോഗത്തില് കണ്ടെയ്ന്മെന്റ് സോണ് നിയന്ത്രണങ്ങളും ചര്ച്ചയായി. മുക്കം എംഎല്എ ഓഫീസിലാണ് യോഗം നടന്നത്. നിയന്ത്രണങ്ങളും ജാഗ്രതയും ശക്തമായി തുടരുന്നതിനും ബോധവത്കരണ പ്രവര്ത്തനങ്ങള് പ്രാദേശിക തലത്തിലും സംസ്ഥാന തലത്തിലും നടത്തുതുന്നതിനുംതിനും തീരുമാനമായി. മുക്കം നഗരസഭ ചെയര്മാന് പി.ടി. ബാബു, കൊടിയത്തൂര് പഞ്ചായത്ത് പ്രസിഡന്റ് ഷംലൂലത്ത് വിളക്കോട്ടില്, കാരശേരി പഞ്ചായത്ത് വൈസ് പ്രസിഡന്റ് ആമിന ഏടത്തില്, ഡിവൈഎസ്പി ടി.കെ. അഷ്റഫ്, മുക്കം ഇന്സ്പെക്ടര് കെ.പി. അഭിലാഷ് തുടങ്ങിയവരും യോഗത്തില് പങ്കെടുത്തു. | malyali |
তৃণমূলের ওয়ার্কিং কমিটিতে গোয়া, ত্রিপুরা, মেঘালয়ের কেউ নেই, এর অর্থ কী? দ্য ওয়াল ব্যুরো: একুশের ভোটে বাংলায় বিজেপিকে ঠেকানোর পর গত আটনমাসে তৃণমূলের সারা ভারতে বিস্তারের আকাঙ্খা বারবার করে দেখা গিয়েছে প্রাথমিক ভাবে বাংলার বাইরে যে দুটি রাজ্যে তৃণমূল কংগ্রেস ফুটপ্রিন্ট রাখতে চেয়েছে তার মধ্যে রয়েছে গোয়া এবং ত্রিপুরা পরে দেখা গিয়েছে, মেঘালয়ের প্রাক্তন মুখ্যমন্ত্রী মুকুল সাংমার নেতৃত্বে এক ডজন কংগ্রেস বিধায়ক যোগ দিয়েছেন তৃণমূলে কিন্তু শনি সন্ধ্যায় কালীঘাটে বৈঠকের পর মমতা বন্দ্যোপাধ্যায় যে জাতীয় কর্মসমিতি ঘোষণা করেছেন তাতে গোয়া, ত্রিপুরা বা মেঘালয়ের কেউ নেই এক মাত্র উত্তরপ্রদেশ থেকে একজন প্রতিনিধি রয়েছেন যে উত্তরপ্রদেশে সম্প্রতি দিদি প্রচারে গিয়েছিলেন আর রয়েছেন প্রাক্তন কেন্দ্রীয় অর্থমন্ত্রী যশবন্ত সিনহা অটলবিহারী বাজপেয়ী সরকারের অর্থমন্ত্রী যশবন্ত সিনহা তৃণমূলে যোগ দেওয়ার পর তাঁকে সর্বভারতীয় সহ সভাপতি করা হয়েছিল তেমনই গোয়ার প্রাক্তন কংগ্রেসি মুখ্যমন্ত্রী লুইজিনহো ফেলেইরো তৃণমূলে যোগ দেওয়ার পর রাজ্যসভার সাংসদ করার পাশাপাশি সর্বভারতীয় সহ সভাপতি পদ দেওয়া হয়েছিল তাঁকে এদিন যে কর্মসমিতি ঘোষণা করেছেন দিদি তাতে যশবন্ত থাকলেও নেই লুইজিনহো প্রথা হল, জাতীয় কর্মসমিতির সদস্যরাই কেবল সর্বভারতীয় পদ পান অন্তত সাবেক কংগ্রেসে তেমনই চল রয়েছে অনেকের বক্তব্য, লুইজিনহোকে তৃণমূলে আনার পিছনে প্রশান্ত কিশোর অনুঘটকের কাজ করেছিলেন তাতে অভিষেকেরও সঙ্গত ছিল তা ছাড়া ত্রিপুরাতেও পুর নির্বাচনে যে আগ্রাসী মুড নিয়ে তৃণমূল ঝাঁপিয়েছিল তাতে টিম অভিষেক ও আই প্যাকের ভূমিকাই ছিল মূল সেখান থেকেও কাউকে কর্মসমিতিতে নেননি দিদি এমনকি সুস্মিতা দেবও নন সর্বোপরি মেঘালয়ে মুকুল সাংমাদের তৃণমূলে আনার নেপথ্যে ছিলেন প্রশান্ত কিশোর সামগ্রিক ভাবে এই গোটা বিষয়টিকে তৃণমূলের অনেকে দলের অভ্যন্তরীণ ক্ষমতার সিঁড়িতে অভিষেকের উত্থানপতনের সঙ্গে জুড়ে দেখতে চাইছেন কারণ, এও দেখা যাচ্ছে ডেরেক ও ব্রায়েন বা সৌগত রায়ের মতো অভিষেক ঘনিষ্ঠ নেতারাও ওয়ার্কিং কমিটিতে জায়গা পাননি তবে পর্যবেক্ষকদের একাংশের এও বক্তব্য, এই ব্যাখ্যা অ্যাবসোলিউট বা চূড়ান্ত নয় এর থেকে প্রাথমিক ভাবে স্পষ্ট যে প্রশান্ত কিশোর ও আইপ্যাকের উপর দিদি অসন্তুষ্ট তাই আপাতত গোয়া, ত্রিপুরা বা মেঘালয় নিয়ে তাঁর কোনও উত্সাহই নেই এও ইঙ্গিত পাওয়া যাচ্ছে যে অভিষেকের হালফিলের ভূমিকা নিয়ে দিদি হয়তো খুশি নয় কিন্তু এও মনে রাখতে হবে, দিদির মুড রৌদ্রছায়ার মতো সুতরাং আপাতত নবীনপ্রবীণে সমাবেশে দলে ভারসাম্য রাখতে চাইলেন দিদি কিন্তু এর পরেও ঘরোয়া আলোচনাতেও দলের কোনও নেতা কি জোরের সঙ্গে বলতে পারবেন যে অভিষেক নম্বর টু নয়? সুতরাং পরিস্থিতি কিছুটা শীতল হলে দিদি কাকে কোন পদ দেন, সেটাও এখন দেখার | bengali |
#teccc_options {
padding-top: 1em;
clear: both;
}
.teccc_options_col1 {
float: left;
position: relative;
width: 160px;
padding-left: 10px;
clear: left;
}
.teccc_options_col2, .teccc_options_col2 p {
width: 80%;
}
.teccc_options_col2 {
padding-left: 205px;
padding-bottom: 15px;
}
.teccc_options_col2 p {
color: #666;
margin-left: 1em;
}
table.teccc.form-table td {
vertical-align: top;
} | code |
Terrorist Arrest : నవరాత్రులో ఆర్డీఎక్స్ పేలుడుకు ప్లాన్.. నలుగురు అరెస్ట్ మరో నెల రోజుల్లో ఉత్తరాధిన నవరాత్రుల పూజలు అంగరంగ వైభవంగా కొనసాగనున్నాయి. అయితే ఈ సంధర్భంలోనే ఉగ్రవాదులు భారీ పేళుళ్లకు స్కెచ్ వేశారు. నవరాత్రుల సమయంలో రామ్లీలా మైదానంతో పాటు దుర్గా పూజా మండపాల వద్ద పేలుళ్లకు ఉగ్రవాదులు కుట్ర పన్నిన నలుగురు ఉగ్రవాదులను ఢిల్లీ పోలీసులు అరెస్ట్ చేశారు. కాగా అరెస్ట్ చేసిన ఉగ్రవాదులు పాకిస్థాన్లోని ఫామ్హౌస్లో శిక్షణ పొందారని, ఆర్డీఎక్స్ బాంబును అండర్వరల్డ్ సాయంతో ఢిల్లీకి తీసుకువచ్చారని కమిషనర్ ఠాకూర్ తెలిపారు. కాగా 1993 తర్వాత ఆర్డీఎక్స్ బాంబును రాజధానికి తరలించడం ఇదే ప్రథమంగా ఆయన తెలిపారు.. దావూద్ ఇబ్రహీం సోదరుడు అనీస్ ఇబ్రహీం ఆర్డీఎక్స్ బాంబును భారత్కు తరలించడంలో కీలకపాత్ర పోషించాడు. ఆరుగురు ఉగ్రవాదుల్లో ఒసామా, జీషాన్కు 15 రోజుల శిక్షణ కూడా అనీస్ ఇబ్రహీం ఇప్పించాడని ఠాకూర్ తెలిపారు. దర్యాప్తు కొనసాగుతోందని చెప్పారు. | telegu |
ಎಸ್ಪಿಬಿ ಸಾವು ಇಡೀ ಭಾರತೀಯ ಚಿತ್ರರಂಗದಲ್ಲಿ ಶೂನ್ಯಭಾವ ಸೃಷ್ಟಿಸಿದೆ: ಬಿ. ಶ್ರೀರಾಮುಲು ಬೆಂಗಳೂರು: ಗಾನ ಗಂಧರ್ವ ಎಸ್ಪಿ ಬಾಲಸುಬ್ರಣ್ಯಂ ಅವರ ನಿಧನಕ್ಕೆ ರಾಜ್ಯ ಆರೋಗ್ಯ ಸಚಿವ ಬಿ. ಶ್ರೀರಾಮುಲು ಕಂಬನಿ ಮಿಡಿದಿದ್ದಾರೆ. ಈ ಬಗ್ಗೆ ಟ್ವೀಟ್ ಮಾಡಿರುವ ಅವರು. ದೇಶ ಕಂಡ ಶ್ರೇಷ್ಠ ಗಾಯಕ, ಹಲವಾರು ಭಾಷೆಗಳಲ್ಲಿ ಹಾಡಿ ಕೋಟ್ಯಂತರ ಅಭಿಮಾನಿಗಳ ಪ್ರೀತಿ ಸಂಪಾದಿಸಿದ ಹಾಡುಗಾರ, ನಟ, ಸಂಗೀತ ನಿರ್ದೇಶಕ ಶ್ರೀ ಎಸ್ ಪಿ ಬಾಲಸುಬ್ರಹ್ಮಣ್ಯಂ ಅವರು ನಮ್ಮನ್ನೆಲ್ಲ ಅಗಲಿದ್ದಾರೆ. ಕೊರೊನ ಸೋಂಕಿನೊಂದಿಗೆ ಸತತವಾಗಿ ಹೋರಾಡಿದ ಅವರು ಕೊನೆಗೂ ವಿಧಿಯ ಕರೆಗೆ ಓಗೊಟ್ಟು ಗಾಯನ ನಿಲ್ಲಿಸಿ ಚಿರಮೌನಕ್ಕೆ ಶರಣಾಗಿದ್ದಾರೆ. ಅವರ ಸಾವು ಇಡೀ ಭಾರತೀಯ ಚಿತ್ರರಂಗದಲ್ಲಿ ಶೂನ್ಯಭಾವ ಸೃಷ್ಟಿಸಿದೆ. ಎಲ್ಲರ ಪ್ರೀತಿಯ ಬಾಲು ಅವರ ಅಗಲಿಕೆ ಎಂದೂ ತುಂಬಲಾರದ ನಷ್ಟ. ಅವರ ಕುಟುಂಬಕ್ಕೆ, ಅಭಿಮಾನಿಗಳಿಗೆ ಈ ನೋವು ತಡೆದುಕೊಳ್ಳುವ ಶಕ್ತಿ ಭಗವಂತ ನೀಡಲಿ. ನಮ್ಮಂತಹ ಎಷ್ಟೋ ಅಭಿಮಾನಿಗಳ ಹೃದಯದಲ್ಲಿ ಎಸ್ ಪಿ ಬಿ ಎಂದಿಗೂ ಅಜರಾಮರ ಎಂದಿದ್ದಾರೆ. | kannad |
کوالالمپور اردو پوائنٹ اخبارتازہ ترین اے پی پی 09 جون2017ء ملائشین پام ائل کے نرخ کم ہو گئےبرسا ملائشیاء ڈرائیویٹیو ایکس چینج میں اگست کے لئے پام ائل کی سپلائی مسلسل دوسرے سیشن گراوٹ کے ساتھ 2444 رنگیٹ57324 ڈالرفی ٹن طے پائی منڈی میں پام ائل کی پچیس پچیس ٹن کی 46998 لاٹس کا کاروبار ہوا | urdu |
UP Election: रूठे हुए अरुण सिंह को मनाने में कामयाब हुईं राज्यमंत्री, क्या सदर विधानसभा के इतिहास को बदलने में भी मिलेगी जीत उत्तर प्रदेश में विधानसभा चुनाव Uttar Pradesh Assembly Election 2022 के लिए आज तीसरे चरण का मतदान जारी है. वहीं भाजपा BJP में अरुण सिंह Arun Singh एक ऐसा जाना पहचाना नाम है जिसने भाजपा के दुर्दिन में भाजपा की अलख जगाए रखी. लेकिन भाजपा के जब अच्छे दिन आए तब इन्होंने सपना देखा था जनप्रतिनिधि बनने का. लेकिन टिकट नहीं मिलने के कारण खुलकर विरोध कर दिया जिसके कारण उन्हें भाजपा से बाहर होना पड़ा. ऐसे में आज वह एक हत्या के मामले में जेल में निरूद्ध है. लेकिन चल रहे विधानसभा चुनाव के मद्देनजर भाजपा से निकाले गए अरुण सिंह की शरण में भाजपा की गाजीपुर सदर Ghazipur Sadar विधायक और सहकारिता राज्यमंत्री को जाना पड़ा और उन्होंने अरुण सिंह की पत्नी से इस चुनाव में समर्थन मांगा. जिसके बाद उन्होंने अपना बड़ा मन करते हुए समर्थन देने का वादा किया. जिला सहकारी बैंक के पूर्व अध्यक्ष अरूण कुमार सिंह की पत्नी शीला सिंह ने भाजपा गाजीपुर सदर प्रत्याशी डॉ संगीता बलवंत के सर पर अपना हाथ रख कर उनको जीत का आशीर्वाद दिया और कहा कि प्रधानमंत्री नरेन्द्र मोदी और मुख्यमंत्री योगी आदित्यनाथ के विजय ध्वज को मजबूत करने के लिए मैं अपने समस्त सहयोगियों के साथ भाजपा प्रत्याशी डा संगीता बलवंत को भारी जीत के लिए तन मन से सहयोग प्रदान करूंगी. 2014 के लोकसभा चुनाव में टिकट के प्रबल दावेदार थे अरुण सिंह संगीता बलवंत ने कहा कि अरुण सिंह जी एक बहुत बड़े समाज सेवी और संघर्षशील व्यक्तित्व के धनी व्यक्ति हैं. उनकी पत्नी के आज इस आशीर्वाद से भाजपा की जीत दोगुनी हो गयी है और यह खुला समर्थन यह सिद्ध करता है की भाजपा की 2022 की यह जीत 2017 से भी बुलंद होगी. इसके लिए मैं हृदय से धन्यवाद आभार प्रकट करती हूं. बताते चलें कि पूर्व भाजपा नेता अरुण सिंह 2014 के लोकसभा चुनाव में टिकट के प्रबल दावेदार में थे. लेकिन उन्हें टिकट नहीं मिला और उन्होंने पार्टी से बगावत कर ली, इस बगावत के कारण पार्टी ने उन्हें बाहर का रास्ता दिखा दिया. उसके पश्चात पत्नी शीला सिंह ने भी जिला पंचायत का चुनाव लड़ा था. वहीं 2022 विधानसभा चुनाव में अपने समर्थकों के कहने पर सदर विधानसभा से नामांकन किया था लेकिन किसी कारण बस उनका नामांकन खारिज हो गया. इस सीट से कोई भी लगातार दो बार नहीं बन पाया विधायक ऐसे में सदर विधानसभा की बात करें तो यहां का इतिहास रहा है की इस सीट से कोई भी लगातार दो बार विधायक नहीं बन पाया है. लेकिन भारतीय जनता पार्टी ने इस मिथ्या को तोड़ने के लिए सहकारिता राज्यमंत्री पर दांव खेला है. वहीं सहकारिता राज्य मंत्री के द्वारा भी रूठे हुए अरुण सिंह के परिवार और उनके समर्थकों के मनाने के इस क्रम से आने वाली राजनीतिक फिजा कितना बदल पाएगी यह तो वक्त ही बता पाएगा. इस अवसर पर प्रवासी अरुण कुमार पांडेय, मनोज बिंद, गोपाल राय, भानु जायसवाल, रविन्द्र जायसवाल, प्रकाश केशरी, शिवप्रसाद सिंह, दीपक सिंह, अमरेश यादव, सुदामा बिंद,अनुप खरवार सहित आदि अन्य लोग उपस्थित रहे. UP Election: उन्नाव में PM मोदी ने अखिलेश यादव पर साधा निशाना, कहा सीएम उम्मीदवार की सीट ही मुश्किल में फंसी Uttar Pradesh Election: बिहार NDA में सहयोगी पार्टी VIP हुई योगी सरकार पर हमलावर, लगाया जनता के शोषण का आरोप | hindi |
અભિનેતા અને પ્રોડ્યુસર સચિન જોશી સામે મની લૉન્ડરિંગનો કેસ બનતો જ નથી મુંબઈ પી.ટી.આઇ. : અહીંની સ્પેશ્યલ કોર્ટે નોંધ્યું હતું કે શહેરની ઓમકાર રિયલ્ટર્સ ઍન્ડ ડેવલપર્સ ફર્મ સામેના મની લૉન્ડરિંગ કેસમાં અભિનેતાપ્રોડ્યુસર સચિન જોશીની સીધી કે આડકતરી સંડોવણી નથી.સ્પેશ્યલ પીએમએલએ જજ એમ. જી. દેશપાંડેએ સોમવારે સચિનના જામીન મંજૂર કર્યા હતા. ઈડીએ ઉપરોક્ત કેસમાં ગયા વર્ષે ૧૪ ફેબ્રુઆરીએ સચિનની ધરપકડ કરી હતી. ૩૭ વર્ષનો અભિનેતા હાલ સુપ્રીમ કોર્ટે તબીબી મેડિકલ કારણોસર મંજૂર કરેલા વચગાળાના જામીન પર બહાર છે. અદાલતે આદેશમાં નોંધ્યું હતું કે આરોપી સામે મની લૉન્ડરિંગનો કેસ બનતો હોય એમ જણાતું નથી.ઉલ્લેખનીય છે કે સચિન જોશી જેએમજે ગ્રુપના પ્રમોટર અને બિઝનેસમૅન જે. એમ. જોશીનો પુત્ર છે જે ગુટકા અને પાનમાસાલાના ઉત્પાદન અને હૉસ્પિટૅલિટીના વ્યવસાયમાં કાર્યરત છે. સચિને જૅકપોટ સહિત કેટલીક ફિલ્મોમાં અભિનય કર્યો છે અને કેટલીક ફિલ્મોનું નિર્માણ પણ કર્યું છે.સચિન ઉપરાંત ઓમકાર રિયલ્ટર્સ ઍન્ડ ડેવલપર્સના ચૅરમૅન કમલ કિશોર ગુપ્તા અને ફર્મના મૅનેજિંગ ડિરેક્ટર બાબુલાલ વર્મા પણ આ કેસના આરોપી છે. બન્ને આરોપીઓ હાલ જુડિશ્યલ કસ્ટડીમાં છે. | gujurati |
என்னால இனி வாழவே முடியாது குடிகார கணவரால், கர்ப்பிணி பெண் எடுத்த விபரீத முடிவு.. 38 வயதான செல்வகுமார் என்ற நபர், ராணிப்பேட்டை சோளிங்கர் அடுத்த கொடைக்கல் கிராமத்தில் வசித்து வந்துள்ளார். கூலித்தொழில் செய்து வரும் இவருக்கு 30 வயதான ஈஸ்வரி என்ற மனைவியும், 5 வயதான ஒரு மகளும், 3 வயதான ஒரு மகனும் உள்ளனர். ஈஸ்வரி தற்போது மீண்டும் கர்ப்பிணியாக உள்ள நிலையில், செல்வகுமார் தினமும் குடித்து விட்டு, வேலைக்கு செல்லாமல் இருந்துள்ளார். மேலும், வீட்டில் இருக்கு செல்வகுமார் தனது மனைவியுடன் தினமும் தகராறு செய்து வந்துள்ளார். இதனால் இருவருக்குமிடையே அடிக்கடி குடும்ப தகராறு ஏற்பட்டு வந்துள்ளது. இதனால் மனமுடைந்த 3 மாத கர்ப்பிணியான ஈஸ்வரி வீட்டில் தூக்கிட்டு தற்கொலை செய்து கொண்டார். இதுபற்றி தகவலறிந்த கொண்டபாளையம் போலீசார் ஈஸ்வரியின் சடலத்தை மீட்டு சோளிங்கர் அரசு மருத்துவமனைக்கு பிரேத பரிசோதனைக்காக அனுப்பி வைத்தனர். இச்சம்பவம் குறித்து போலீசார் வழக்கு பதிவு செய்து விசாரணை நடத்தி வருகின்றனர். ALSO READ: இன்று மொத்தம் 12 மாவட்டங்களுக்கு கனமழை எச்சரிக்கை: வானிலை ஆய்வு மையம் | tamil |
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ವಿಜಯೇಂದ್ರ ಸೂಪರ್ ಸಿಎಂ ಅಲ್ಲ: ಬಿ.ಎಸ್.ಯಡಿಯೂರಪ್ಪ ಬೆಂಗಳೂರು: ತಮ್ಮ ಪುತ್ರ ಬಿ.ವೈ.ವಿಜಯೇಂದ್ರ ಸೂಪರ್ ಸಿಎಂ ಎಂದು ವಿರೋಧ ಪಕ್ಷದ ನಾಯಕ ಸಿದ್ದರಾಮಯ್ಯ ಅವರ ಆರೋಪದಲ್ಲಿ ಹುರುಳಿಲ್ಲ. ವಿಜಯೇಂದ್ರ ಸೂಪರ್ ಸಿಎಂ ಅಲ್ಲ ಎಂದು ಬಿ.ಎಸ್.ಯಡಿಯೂರಪ್ಪ ಹೇಳಿದರು. ದೆಹಲಿಯಲ್ಲಿ ಕರ್ನಾಟಕ ಭವನದ ಕಾವೇರಿ ಕಟ್ಟಡದ ನಿರ್ಮಾಣ ಕಾಮಗಾರಿಗೆ ಭೂಮಿ ಪೂಜೆ ಮಾಡಿದ ಬಳಿಕ ಸುದ್ದಿಗಾರರ ಪ್ರಶ್ನೆಗೆ ಉತ್ತರಿಸಿ, ವಿಜಯೇಂದ್ರ ಬಿಜೆಪಿ ರಾಜ್ಯ ಘಟಕದ ಉಪಾಧ್ಯಕ್ಷರಾಗಿದ್ದಾರೆ. ರಾಜ್ಯದಲ್ಲಿ ಪಕ್ಷವನ್ನು ಸಂಘಟಿಸುವ ಕಾರ್ಯ ನಿರ್ವಹಿಸುವ ಕೆಲಸ ಮಾಡುತ್ತಿದ್ದಾರೆಯೇ ಹೊರತು, ರಾಜ್ಯ ಸರ್ಕಾರದ ಯಾವುದೇ ಚಟುವಟಿಕೆಗಳಲ್ಲಿ ಹಸ್ತಕ್ಷೇಪ ಮಾಡುತ್ತಿಲ್ಲ ಎಂದರು. ಜಾತ್ಯತೀತ ಜನತಾದಳದ ಮುಖಂಡ ಎಚ್.ಡಿ.ಕುಮಾರಸ್ವಾಮಿ ಮತ್ತು ತಮ್ಮ ಭೇಟಿಯ ಕುರಿತ ಪ್ರಶ್ನೆಗೆ ಉತ್ತರಿಸಿದ ಅವರು, ಕುಮಾರಸ್ವಾಮಿ ವಿರೋಧ ಪಕ್ಷದ ಪ್ರಮುಖ ನಾಯಕರು. ತಮ್ಮ ಕ್ಷೇತ್ರದ ಅಭಿವೃದ್ಧಿಗೆ ಸಂಬಂಧಿಸಿದಂತೆ ಚರ್ಚಿಸಲು ಬಂದಿದ್ದರು. ಆ ಬಗ್ಗೆ ಅನಗತ್ಯ ಊಹಾಪೋಹಗಳು ಬೇಡ ಎಂದು ಹೇಳಿದರು. ನೂತನ ಕಟ್ಟಡದಲ್ಲಿ ಎರಡು ತಳ ಮಹಡಿ, ನೆಲ ಮಹಡಿ ಮತ್ತು ಆರು ಮಹಡಿಗಳು ಸೇರಿ ಒಂಭತ್ತು ಮಹಡಿಗಳು ಇರಲಿವೆ. ಮೆಸರ್ಸ್ ಬಾಲಾಜಿ ಕೃಪಾ ಪ್ರಾಜೆಕ್ಟ್ಸ್ ಪ್ರೈವೇಟ್ ಲಿಮಿಟೆಡ್ ಅವರಿಗೆ ನಿರ್ಮಾಣ ಕಾರ್ಯ ವಹಿಸಲಾಗಿದೆ. ಮುಂದಿನ 24 ತಿಂಗಳಲ್ಲಿ ಕಾಮಗಾರಿ ಪೂರ್ಣಗೊಳ್ಳಲಿದೆ ಎಂದು ಯಡಿಯೂರಪ್ಪ ಹೇಳಿದರು. | kannad |
Partially Broker owned! Professional office building being constructed on E. Barnett Road. East of Foster Denman Offices. Estimated completion June/July 2019. Owner reserves the right to approve tenant and use. West end cap space.
$2.00/sqft/mo Net 3. Minimum 5 year lease. | english |
વેસ્ટ ઈન્ડિઝના પ્રવાસ દરમિયાન ઘૂંટણની ઈજાથી પરેશાન બેન સ્ટોક્સ લંડનઃ ઈંગ્લેન્ડનો ઓલરાઉન્ડર બેન સ્ટોક્સ ઈજાની સમસ્યા સામે ઝઝૂમી રહ્યો છે. 30 વર્ષીય ક્રિકેટરે કહ્યું કે વેસ્ટ ઈન્ડિઝના પ્રવાસ દરમિયાન તેને ઘૂંટણમાં ઈજા થઈ હતી, જેનું સીટી સ્કેન કરવાનું બાકી છે. જો રૂટની આગેવાની હેઠળ, ઈંગ્લેન્ડ વેસ્ટ ઈન્ડિઝ સામે ત્રણ મેચની શ્રેણી હારી ગયું હતું, જેમાં વેસ્ટ ઈન્ડિઝે ગ્રેનાડામાં છેલ્લી મેચ 10 વિકેટથી જીતી હતી. આ દરમિયાન સ્ટોક્સે શ્રેણીમાં 99 ઓવર ફેંકી હતી. Advertisement DailyMail.co.uk ના એક અહેવાલ મુજબ, બુધવારે સ્ટોક્સે તેના પોડકાસ્ટમાં કહ્યું હતું કે જ્યાં સુધી સીટી સ્કેન દ્વારા ઈજાની ગંભીરતા જાહેર ન થાય ત્યાં સુધી તે આ રમત વિશે વધુ કોઈ યોજના બનાવશે નહીં. આ પણ વાંચો: IPL 2022: રવિ શાસ્ત્રીએ શુભમન ગિલને સૌથી પ્રતિભાશાળી ક્રિકેટર તરીકે ઓળખાવ્યો, જોસ બટલરે પ્રખ્યાત કૃષ્ણની આગાહી કરી ઈંગ્લેન્ડના વાઇસકેપ્ટનને પણ એશિઝ દરમિયાન બાજુમાં તણાવ થયો હતો અને વેસ્ટ ઈન્ડિઝ સામેની શરૂઆતની ટેસ્ટમાં બોલિંગ કરવાની અપેક્ષા ન હતી, પરંતુ ત્રણ ટેસ્ટમાં તેણે 99 ઓવર ફેંકી હતી, જે કોઈપણ ઝડપી બોલર દ્વારા સૌથી વધુ છે. સૌથી વધુ ઓવરોમાંની એક ગયો ઇંગ્લેન્ડની આગામી સિઝન 2 જૂને ન્યૂઝીલેન્ડ સામે લોર્ડ્સમાં શરૂ થશે. પ્રથમ પ્રકાશિત:6 એપ્રિલ, 2022, સાંજે 6:15 કલાકે Advertisement પ્રથમ પ્રકાશિત:6 એપ્રિલ, 2022, સાંજે 6:15 કલાકે | gujurati |
In the year 2044, reality is an ugly place. The only time teenage Wade Watts really feels alive is when he's jacked into the virtual utopia known as the OASIS. Wade's devoted his life to studying the puzzles hidden within this world's digital confines--puzzles that are based on their creator's obsession with the pop culture of decades past and that promise massive power and fortune to whoever can unlock them.
Given the success of the book, it wasn't long before it was optioned for a movie, though it took some time before production began. Once famed director and movie-maker Steven Spielberg signed on, however, excitement truly filled the air in hopes that the wonderful world that Cline created would translate well to the silver screen.
Rush fans, of course, wondered whether or not 2112 would be featured at all in the film. Well, that question may have not been answered yet, but if the movie's official trailer, which was just released today, is any indication, Rush's music will play a part. Tom Sawyer is featured heavily in the trailer, which can be viewed below, or directly at this LINK.
For more information about the movie, which is slated for a 2018 release, please click HERE.
We'll have more news on the film, and any other inclusion of Rush, as it becomes available.
Happy 64th Birthday, Geddy Lee! | english |
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गांवों में चुनाव की सरगर्मी तेज हो रही चंदौली । विभिन्न दलों के कार्यकर्ता अब गांवों में आमनेसामने अपनी पार्टी के प्रत्याशी की जीत सुनिश्चित करने में जुटे हैं। गांवगांव वोटरों को लुभावने काम किया जा रहा है।Click here to get the latest updates on State Elections 2022 कई गांवों में हैंडपंप ,सोलर लाइट एवं धन देने का लालच कार्य जारी है ।यही नहीं कुछ जगह झंडे लगाने उखाड़ने के विवाद भी शुरू हो गया है जिससे कार्यकर्ता आमनेसामने आ जा रहे हैं । शिकायत होने पर पुलिस सक्रिय होकर मामले को सुलझा रही है। धीना पुलिस का कहना है की कुछ गांवों में इस तरह की शिकायत आयी थी जिसका निदान कर दिया गया है।वहीं लोगो को निर्भीक होकर मतदान करने को प्रेरित किया जा रहा है।चुनाव में खलल करने वाले बख्शे नहीं जाएंगे। The post गांवों में चुनाव की सरगर्मी तेज हो रही appeared first on Breaking news Latest news News in hindi Tarun Mitra तरुण मित्र. | hindi |
نیہ شرما چھِ اَکھ ہِندوستٲنؠ اَداکارہ یۄس فِلمَن مَنٛز چھِ کٲم کَران.
زٲتی زِندگی
فِلمی دور
== حَوالہٕ == | kashmiri |
ઉત્તર ગુજરાતમાં ભાજપના આગેવાન વીજ ચોરી કરતા પકડાયા, MGVCLએ કર્યો 10 લાખનો દંડ રાજ્યમાં ઘણી વખત વીજ ચોરીના કિસ્સાઓ સામે આવે છે અને ક્યારેક તો નેતાઓના પરિવારના સભ્યો જ વીજ ચોરી કરતા પકડાયા હોવાના મામલાઓ સામે આવે છે. ત્યારે આવી જ એક ઘટના આણંદમાં સામે આવી છે. આણંદ નગરપાલિકાના પૂર્વ કોર્પોરેટર પોતાના ધંધાકીય એકમ પર વીજ ચોરી કરતા હોવાની માહિતી વડોદરા વિજિલન્સને મળી હતી. તેથી વીજ કંપનીના અધિકારીઓ દ્વારા પૂર્વ કોર્પોરેટરના કોમ્પલેક્ષમાં રેડ કરવામાં આવી હતી અને ત્યારે જાણવા મળ્યું હતું કે, આ કોમ્પલેક્સમાં પાંચમાંથી બે કનેક્શન ડાયરેકટ આપવામાં આવ્યા છે. તેથી અધિકારીઓ દ્વારા આ સમગ્ર મામલે કાર્યવાહી કરવામાં આવી છે. રિપોર્ટ અનુસાર આણંદ નગરપાલિકાના પૂર્વ કોર્પોરેટર હિમેશ પટેલ દ્વારા તેમના ધંધાકીય એકમ હેપી હોમ્સમાં વીજચોરી થઈ રહી હોવાની વાત વડોદરા વીજ કંપનીની વિજિલન્સ ટીમને મળી હતી. તેથી વિજિલન્સની ટીમ દ્વારા આ બાબતે હેપી હોમ્સમાં દરોડો પાડવામાં આવ્યો હતો અને દરોડા દરમિયાન જાણવા મળ્યું હતું કે હેપી હોમ્સની અંદર પાંચ કનેક્શન હતા અને આ પાંચ કનેક્શનમાંથી બે કનેક્શન ડાયરેક્ટ આપવામાં આવ્યા હતા. એટલે કે બે કનેક્શનમાં સીધું જોડાણ આપીને વીજ ચોરી કરવામાં આવતી હતી. આ સમગ્ર મામલે વીજ કંપનીના કર્મચારીઓ દ્વારા ભાજપના નેતા હિમેશ પટેલને 10.70 લાખ રૂપિયાનો દંડ ફટકાર્યો હતો અને આ સમગ્ર મામલે વિજિલન્સની ટીમ દ્વારા પોલીસ સ્ટેશનમાં ફરિયાદ નોંધાવવામાં આવી હતી. ભાજપના જ નેતા વીજ ચોરી કરતા પકડાયા હોવાના કારણે જિલ્લા ભાજપ સંગઠનમાં પણ અને તર્ક વિતર્કો થઇ રહ્યા છે. એક તો ભાજપના નેતા વીજ ચોરી કરતા પકડાયા અને ત્યારબાદ જ્યારે વીજ કંપની દ્વારા તેમને દંડ કરવામાં આવ્યો ત્યારે ભાજપના નેતાએ મધ્ય ગુજરાત વીજ કંપની સામે દંડની રકમમાં વાંધો લીધો હતો અને આ બાબતે તેમને જરૂરી કાર્યવાહી પણ હાથ ધરી છે. મહત્ત્વની વાત છે કે, થોડા સમય પહેલા આવો જ એક કિસ્સો રાજકોટમાં પ્રકાશમાં આવ્યો હતો. દૂધસાગર રોડ પર આવેલા શિવાજી નગરમાં રહેતા અશ્વિન રાઠોડ નામના વ્યક્તિએ ડાયરેક્ટ પાવર સપ્લાય કરી હોવાના કારણે વીજકંપનીના કર્મચારીઓ દ્વારા તેના ઘરનું કનેકશન કાપી નાંખ્યું હતું. તેથી રાજકોટ મહાનગરપાલિકાના વિપક્ષના નેતાના દીકરા અને અશ્ચીન રાઠોડ દ્વારા વીજ વિભાગના કર્મચારી પર હુમલો કર્યો હતો અને તેમને મારી નાખવાની ધમકી આપી હતી. | gujurati |
பால் பொருள்கள் ஏற்றுமதி 3.2 சரிவு: மாநிலங்களவையில் அமைச்சா் தகவல் இந்தியாவில் கடந்த ஆண்டு ஏப்ரல் முதல் அக்டோபா் வரையிலான காலகட்டத்தில் பால் பொருள்கள் ஏற்றுமதி 3.2 சதவீதம் சரிந்துவிட்டது என்று மாநிலங்களவையில் வெள்ளிக்கிழமை தெரிவிக்கப்பட்டது. இது தொடா்பாக மத்திய மீன்வளம், கால்நடை பராமரிப்பு மற்றும் பால் வளத்துறை இணையமைச்சா் சஞ்சீவ் குமாா் பால்யான் எழுத்து மூலம் அளித்த பதிலில் கூறியிருப்பதாவது: கடந்த ஆண்டு ஏப்ரல் முதல் அக்டோபா் வரையிலான காலகட்டத்தில் பால் பொருள்கள் ஏற்றுமதி 3 சதவீதம் குறைந்துவிட்டது ரூபாயின் மதிப்பில். கரோனா தொற்று பிரச்னை காரணமாகவே இந்த சரிவு ஏற்பட்டுள்ளது. வேளாண்மை, உணவுப் பதப்படுத்துதல் ஏற்றுமதி மேம்பாட்டு ஆணையத்துடன் இணைந்து இது தொடா்பாக பல்வேறு திட்டங்களை செயல்படுத்த உள்ளது. இதன் மூலம் பால் பொருள்கள் ஏற்றுமதியில் ஏற்பட்ட தொய்வு சரி செய்யப்படும். பால் பொருள்கள் ஏற்றுமதியில் மட்டுமல்லாது, இறக்குமதியிலும் அந்த குறிப்பிட்ட காலகட்டத்தில் சரிவு ஏற்பட்டது. பண மதிப்பின் அடிப்படையில் மட்டுமல்லாது, இறக்குமதி அளவும் வெகுவாக குறைந்தது என்றாா். எனினும், ஏற்றுமதி எத்தனை கோடி ரூபாய் அளவுக்கு குறைந்தது என்பது தொடா்பான விவரத்தை அமைச்சா் தெரிவிக்கவில்லை. மற்றொரு கேள்விக்கு பதிலளித்த பால்யான், கரோனா தொற்று பிரச்னையின்போது பால் பொருள் விவசாயிகள், பொருளாதார பிரச்னைகளைச் சந்தித்தனா். அப்போது சிறப்பு நடவடிக்கையாக பால் கூட்டுறவு சங்கங்களைச் சோந்த பால் உற்பத்தியில் உள்ள விவசாயிகளுக்கு கிஸான் கடன் அட்டை வழங்கப்பட்டது. ஜனவரி 22ஆம் தேதி வரை 5.72 புதிய கிஸான் கடன் அட்டைகள் வழங்கப்பட்டுள்ளன என்றாா். | tamil |
নজরে শুভেন্দু, কাঁথি পুরসভায় শুভেন্দু ঘনিষ্ঠ ইঞ্জিনিয়ারের কাছ ৪ কোটি! গ্রেফতার শুভেন্দু অধিকারীর ঘনিষ্ঠ কনিষ্ক সামন্ত কলকাতা সারাদিন ADVERTISEMENT আয়ের চেয়ে হিসাব বহির্ভুত সম্পত্তি প্রায় ২ কোটি টাকা আর এর জেরেই রাজ্যের বিরোধী দলনেতা শুভেন্দু অধিকারীর ছোট ভাই সৌম্যেন্দু অধিকারী ঘনিষ্ট কাঁথি পৌরসভার সহকারি ইঞ্জিনিয়ার দিলীপ বেরাকে গ্রেফতার করলো কাঁথি থানার পুলিশউল্লেখ কাঁথি পৌরসভার রাঙ্গামাটি শ্মশানের জমিতে বেআইনী ভাবে স্টল বানিয়ে বিক্রীর অভিযোগে কিছুদিন আগেই এই ইঞ্জিনিয়ারকে গ্রেফতার করেছিলো পুলিশএই মামলায় তিনি জামিনে মুক্ত হয়েছিলেন গত ২৯ জুন কাঁথি শহরের রাঙামাটি শ্মশানে স্টল নির্মাণের দুর্নীতির অভিযোগ তুলে সরব হন কাঁথি পুরসভার বর্তমান পুরপ্রধান সুবল কুমার মান্না কাঁথি থানায় লিখিত অভিযোগ দায়ের করেন অভিযোগের নাম উঠে আসে প্রাক্তন পুরপ্রধান তথা রাজ্যের বিরোধী শুভেন্দু অধিকারীর ভাই সৌম্যেন্দু অধিকারীর এছাড়াও পুরসভার সহকারি ইঞ্জিনিয়ার দিলীপ বেরা, পুরসভার ঠিকাদার সতীনাথ দাস অধিকারীর নামে মামলা রয়েছে পুলিশ তদন্তে নেমে প্রথমে পুরসভার সহকারী ইঞ্জিনিয়ার দিলীপ বেরাকে গ্রেফতার করে এরপর ঠিকাদার সতীনাথ দাস অধিকারীর ম্যানেজার অলক সাহু ও সৌম্যেন্দু অধিকারীর গাড়ির চালক গোপাল সিংকে গ্রেফতার করে যদিও কাঁথি আদালত শর্তসাপেক্ষে চারজনকে জামিনে মুক্তি দেয় যদিও হাইকোর্ট থেকে রক্ষাকবচ পান সৌমেন্দু অধিকারী পুলিশের এক সুত্রের দাবি তারা শ্মশানের জমি বিক্রীর দুর্নীতির তদন্তে নেমে জানতে পারেন তদন্তে নেমে পুলিশ জানতে পারে, অভিযুক্ত ইঞ্জিনিয়ার দিলীপ বেরার সম্পত্তির পরিমাণ ৩ কোটি ১৯ লক্ষ ৫২ হাজার ৯৪৭ টাকা এর পরেই খোঁজ নিয়ে তারা জানেন এই ইঞ্জিনিয়ার গত ১৯৯৭ সালে অধিকারী আমলেই কাঁথি পুরসভার কাজে যোগ দেন সুত্রের থেকে আরো জানা গেছে পুলিশ হেফাজতে থাকা সহকারী ইঞ্জিনিয়ারের ৩০ এপ্রিল ২০২২ সাল পর্যন্ত বেতন বাবদ একাউন্টে ঢুকেছে ৮৯ লক্ষ ৯৫ হাজার ৮৬৯ টাকা অথচ সম্পত্তি ৩ কোটি ছাড়িয়েছেদিলীপ বাবু কি করে নিজের ও স্ত্রীর নামে এতো বিপুল সম্পত্তি বানালেন তা তদন্ত করে দেখছে পুলিশ এদিন ধৃতকে কাঁথি আদালতে তোলা হলে বিচারক জামিন নাকচ করে ১২ দিনের পুলিশি হেফাজতের নির্দেশ দেন বিশেষ আদালতের বিচারক তথা অতিরিক্ত দায়রা জেলা আদালতের বিচারক তূতীয় সুপর্ণা রায় Tags: Kanthi KolkataSaradin SoumenduAdhikari SuvenduAdhikari | bengali |
মুলোর পাতা খেতে পছন্দ করেন? জানুন উপকারিতা মুলো যে সবজিটা সাধারণত শীতকালেই বাজারে পাওয়া গিয়ে থাকে মুলো আমরা অনেকেই ভালো খাই আবার অনেকেই খাইনা আমরা অনেকেই জানি না যে মুলোর পাতার মধ্যে লুকিয়ে আছে এমন কিছু জিনিস যার ফলে অনিদ্রা, ক্লান্তি এই সমস্ত রোগ গুলিকে সারিয়ে তুলতে সক্ষম হয় মুলোর পাতার মধ্যে রয়েছে ক্লোরিন, সোডিয়াম, ফসফেট, আয়রন, ম্যাগনেসিয়াম, ভিটামিন এ, যার ফলে আমাদের অনেক রোগ নিরাময়ে সাহায্য করে এই মূলো পাতা আমরা সাধারণত মূলো খাওয়ার জন্য বাজার থেকে আনি মূলো খাই কিন্তু যে পাতাগুলো থাকে সে গুলোকে আমরা আবর্জনা ভেবে ফেলে দিই, কিন্তু এই পাতাতেই যে এতগুলো গুণ রয়েছে সেটা আমরা অনেকেই জানিনা মুলোর পাতা পেটের অসুখের জন্য বেশ কার্যকরী মূলত তরকারি অথবা মুলোর পরোটা যদি আপনি এই পাতা ব্যবহার করে খান তবে আপনার শরীরে তৈরি হবে রোগ প্রতিরোধের ক্ষমতা যাদের পাইলসের সমস্যায় আছে তারা মুলো পাতার তৈরি সবজি খেলে অনেক স্বস্তি বোধ করতে পারবেন মুলো পাতায় রয়েছে অ্যান্টিব্যাকটেরিয়াল গুন যেগুলি আপনার শরীরের যে কোন জায়গাতে ফোলা ভাব এবং ব্যথা যদি থাকে তা কম করতে সক্ষম হবে মুলো পাতায় থাকে সোডিয়াম যা শরীরের লবণের পরিমাণ কে বেলেন্স করতে সাহায্য করে যাদের নিম্ন রক্তচাপের সমস্যা আছে তাদের ক্ষেত্রে এই পাতা বেশ উপকারী মুলোর পাতায় থাকা এন থেকে নিনজা হৃদয়কে সুস্থ রাখতে সাহায্য করে পাতায় থাকে ফাইবার যা কোষ্ঠকাঠিন্য রোগকে সুস্থ করতে সক্ষম হয় যাদের জন্ডিস আছে তারা যদি মুলো পাতার রস এবং মিশ্রি একসঙ্গে মিশিয়ে খায় তবে তারা সুস্থ হতে পারবেন চুল ঝরা বন্ধ করতে মুলো শাক বেশ কার্যকরী সর্দি লেগে থাকলে মুলো পাতা অথবা মুলো শাক যদি ভেজে খাওয়া যায় তাহলে সর্দি লাগা কমে যায় দাঁতের মাড়ির সমস্যাকে সমাধান করে মুলো এবং মুলো পাতা | bengali |
காதலன் திருமணம் செய்ய மறுத்ததால் இளம்பெண் தூக்கிட்டு தற்கொலை திருவள்ளூர் அருகே இளம் பெண் தன் காதலித்த காதலன் திருமணம் செய்ய மறுத்ததால் தூக்குப்போட்டு தற்கொலை செய்துக்கொண்ட சம்பவம் அப்பகுதியில் சோகத்தை ஏற்படுத்தியுள்ளது.திருவள்ளூரை அடுத்த மப்பேடு அருகே உள்ள நரசமங்கலம் காலனியை சேர்ந்தவர் சுந்தரமூர்த்தி. இவரது மகள் ஜெயஸ்ரீ 21.இவர் பிபிஏ முடித்துவிட்டு சுங்குவார்சத்திரத்தில் உள்ள தனியார் கம்பெனி ஒன்றில் வேலை செய்து வந்துள்ளார். இந்த நிலையில் கடந்த 24 ஆம் தேதியன்று வேலைக்குச் சென்ற ஜெயஸ்ரீ மீண்டும் வீடு திரும்பவில்லை. இதனால் பதறிப்போன அவரது பெற்றோர் பல இடங்களில் தேடி உள்ளனர். ஆனால் அவர் கிடைக்கவில்லை. காணாமல் போன அவரது மகள் ஜெயஸ்ரீ நரசமங்கலம் பகுதியிலுள்ள ஏரிக்கரையில் உள்ள முட்புதரில் தனது துப்பட்டாவால் தூக்குப்போட்டு தற்கொலை செய்து கொண்டிருப்பதாக அக்கம்பக்கத்தினர் தெரிவித்ததையடுத்து அங்கு சென்ற பெற்றோர், மகள் தூக்கில் தொங்குவதை கண்டு அதிர்ச்சியடைந்தனர். அப்போது ஜெயஸ்ரீ அதே பகுதியை சேர்ந்த ஒருவரை காதலித்து வந்ததும் தெரியவந்தது. இந்த நிலையில் தனது மகளின் சாவில் சந்தேகம் இருப்பதாக ஜெயஸ்ரீயின் தாயார் ரேவதி மப்பேடு போலீசில் புகார் செய்தார். சம்பவ இடத்திற்கு சென்ற போலீசார், ஜெயஸ்ரீயின் மீட்டு பிரேத பரிசோதனைக்காக திருவள்ளூர் அரசு மருத்துவமனைக்கு அனுப்பி வைத்து இது சம்பந்தமாக வழக்கு பதிவு செய்து விசாரித்து வருகின்றனர். முதற்கட்ட விசாரணையில் ஜெயஸ்ரீயின் காதலன் அவரை திருமணம் செய்துகொள்ள மறுத்தது தெரியவந்தது. | tamil |
आयुर्वेद स्नातक पाठ्यक्रम में चार नहीं अब तीन परीक्षाएं होंगी भोपाल, डेस्क रिपोर्ट। नेशनल कमीशन फार इंडियन सिस्टम आफ मेडिसिन यानि भारतीय चिकित्सा पद्धति राष्ट्रीय आयोग ने आयुर्वेद स्नातक कोर्स BAMS की परीक्षा व्यवस्था में बदलाव किया है। बीएएमएस में अब तक स्नातक पाठ्यक्रम को पूरा करने के लिए चार परीक्षाएं होती थीं। इनमें पहला, दूसरा और तीसरा एकएक साल और चौथा डेढ़ साल में भोपाल, डेस्क रिपोर्ट। नेशनल कमीशन फार इंडियन सिस्टम आफ मेडिसिन यानि भारतीय चिकित्सा पद्धति राष्ट्रीय आयोग ने आयुर्वेद स्नातक कोर्स BAMS की परीक्षा व्यवस्था में बदलाव किया है। बीएएमएस में अब तक स्नातक पाठ्यक्रम को पूरा करने के लिए चार परीक्षाएं होती थीं। इनमें पहला, दूसरा और तीसरा एकएक साल और चौथा डेढ़ साल में होता था। वही अब बीएएमएस का पूरा कोर्स पहले की तरह अभी भी साढ़े चार साल का रहेगा, लेकिन अब तीन परीक्षाएं होंगी, जो डेढ़डेढ़ साल अंतराल पर ली जाएंगी। 202223 के सत्र से यह नियम लागू हो जाएंगे। यह भी पढ़े.. भिंड : अज्ञात वाहन ने बाइक सवार को कुचला, युवक की मौत भारतीय चिकित्सा पद्धति राष्ट्रीय आयोग ने 16 फरवरी को अधिसूचना जारी कर दी है। इसके मुताबिक इंटर्नशिप पहले की तरह डेढ़ साल की ही होगी। बताया जा रहा है कि एक परीक्षा कम होने से विद्यार्थियों को फायदा होगा। विश्वविद्यालय भी समय पर परीक्षाएं आयोजित करा सकेंगे। अभी परीक्षाएं समय पर नहीं होने की वजह से डिग्री पूरी करने में देरी होती है। इसके साथ ही मूल्यांकन की समस्या भी हल हो जाएगी। | hindi |
سڈنی اردو پوائنٹ اخبارتازہ ترین اے پی پی 06 جنوری2018ء بگ بیش لیگ بی پی ایل میں کل اتوار کو سڈنی تھنڈر اور ایڈیلیڈ سٹرائیکرز کی ٹیمیں پنجہ ازما ہوں گی تفصیلات کے مطابق اسٹریلوی بگ بیش لیگ کے سنسنی خیز مقابلوں کا سلسلہ جاری ہے لیگ کا 20 واں میچ اج سڈنی تھنڈر اور ایڈیلیڈ سٹرائیکرز کے درمیان سڈنی میں کھیلا جائے گا ایڈیلیڈ سٹرائیکرز چھ پوائنٹس کے ساتھ تیسرے جبکہ سڈنی تھنڈر چار پوائنٹس کے ساتھ پانچویں نمبر پر ہے | urdu |
కేక్తో... ఓ సర్ప్రైజ్ గిఫ్ట్ కేక్తో... ఓ సర్ప్రైజ్ గిఫ్ట్ పు ట్టినరోజు, వార్షికోత్సవాలు అయితే కేక్, గిఫ్ట్లు విడివిడిగా తీసుకువెళతాం. రెంటినీ కలిపితే ఎలా ఉంటుంది అనుకున్నారు సృజనకారులు. సాధారణ కేక్ తియ్యటి రుచులను అందిస్తుంది. అయితే ఈ ఫొటోలో కనిపిస్తోన్న కేక్ మాత్రం తీపితోపాటు కానుకనూ ఇస్తుంది. ఇందుకోసం ప్రత్యేకమైన కేక్ స్టాండును రూపొందించారు. దీనిపై మామూలుగానే కేక్ను ఏర్పాటు చేసుకోవచ్చు. దీని కింద ఉండే స్తూపాకార గొట్టం ద్వారా మీ ప్రియమైనవారికి సర్ప్రైజ్గా బహుమతిని అందివ్వొచ్చు. స్టాండుకు కింద ఉన్న మీట సాయంతో కేక్ మధ్య భాగం నుంచి కానుకతో ఉండే ప్రత్యేకమైన గాజు గ్లాసు బయటకు వస్తుంది. ఆ సమయంలో చక్కటి సంగీతమూ వినిపిస్తుంది. ఇందులో చాక్లెట్లు, బిస్కట్లలాంటి వాటితోపాటు ఫోన్, ఇయర్ రింగ్స్, చైన్ లాంటి వాటిని బహుమతిగా ఇవ్వొచ్చు. | telegu |
গ্রামের নাম মুখেই আনা যায় না, লজ্জায় মাথা হেঁট হয়, বদলে দেওয়ার আর্তি বাসিন্দাদের দ্য ওয়াল ব্যুরো: নাম দিয়ে যায় চেনা! এমনই নাম যে মুখে আনতে লজ্জা লাগে কেউ জিজ্ঞেস করলে অস্ফুটে উচ্চারণ করেই কেটে পড়েন গ্রামবাসীরা লজ্জায় কান লাল হয়ে আসে, জিভ আড়ষ্ট হয়ে যায় জনসমক্ষে তো এমন নাম উচ্চারণ করাই দায় নামবিভ্রাটে ভুগছেন সুইডেনের বহু পুরনো একটি গ্রামের বাসিন্দারা ১৫৪৭ সালে তৈরি হয়েছিল এই গ্রাম তখন থেকেই জনপদ গড়ে উঠেছে কত স্মৃতি, ঐতিহাসিক ঘটনা জড়িয়ে আছে এই গ্রামে প্রাকৃতিক সৌন্দর্যেরও তুলনা নেই তাই জন্মস্থান ছেড়ে চলে যাওয়ার প্রশ্নই ওঠে না শুধু নামটুকু বদলে দেওয়ার জন্যই মরিয়া গ্রামবাসীরা সরকারের কাছে আবেদনের পর আবেদন জমা পড়ছে স্থানীয় প্রশাসনের কাছে কাকুতিমিনতি করেছেন যেমন করেই হোক নামটা পরিবর্তন হলেই সব মুশকিল আসান হবে যে নাম নিয়ে এত হইচই, হাস্যকৌতুক হচ্ছে সেটি হল ফাক Fucke হ্যাঁ, ঠিকই পড়ছেন এই গ্রামের নাম এমনই টলটলে জলের লেক, তার পাশেই খাড়া পাহাড়, চারদিক সবুজে ঘেরা, মাঝে ছোট্ট গ্রাম ফাক আসলে গ্রামকে ঘিরে থাকা লেকের নাম এটাই, তাই গ্রামেরও নাম হয়েছে এমন আর এখন এই নাম নিয়েই বিড়ম্বনায় পড়েছেন গ্রামবাসীরা ন্যাশনাল সার্ভে অব সুইডেনের কাছে দরখাস্ত করে গ্রামবাসীরা বলছেন, সুইডিশ ভাষায় গ্রামের নাম ছিল ফোক্কা Fokka, তারপর কেমন করে যেন লোকমুখে ও টা ইউ হয়ে যায় সরকারি খাতাতেও সেই নামই থেকে যায় সোশ্যাল মিডিয়ায় এই গ্রামের নাম লেখা যায় না সেন্সর আটকে দেয় ঠিকানায় গ্রামের নাম লিখতে হলে সমস্যা হয় উচ্চারণ করতে গেলে লজ্জায় পড়তে হয় তাই গ্রামবাসীদের কাতর আর্তি, যত দ্রুত সম্ভব এই গ্রামের নাম বদলে দিতে হবে পড়ুন দ্য ওয়ালের সাহিত্য পত্রিকা সুখপাঠ | bengali |
#include "VertexBuffer.hpp"
#include <algorithm>
/*****************************************************************************/
VertexBuffer* VertexBuffer::create()
{
return new VertexBuffer();
}
/*****************************************************************************/
void VertexBuffer::destroy()
{
delete this;
}
/*****************************************************************************/
VertexBuffer::VertexBuffer()
: mCount(0)
, mVertexSize(0)
, mData(DataBuffer::create())
{
}
/*****************************************************************************/
void VertexBuffer::setNumOfVertices(uint32_t pCount)
{
if
(pCount NEQ mCount)
{
mData->resize(pCount * mVertexSize);
mCount = pCount;
}
}
/*****************************************************************************/
void VertexBuffer::setAttributeValue(ESemantic pSemantic, uint32_t pIndex, const uint8_t* pDataValue, uint32_t pCount)
{
// Find the semantic
auto lItr = std::find_if(mAttributes.begin(), mAttributes.end(), [pSemantic](const VertexBuffer::SAttributeDesc& pDesc) { return pDesc.mSemantic EQ pSemantic; });
if
(lItr NEQ mAttributes.end())
{
uint32_t lAttributeSize = TranslateVertexAttributeFormatToByte(lItr->mFormat);
for (uint32_t i = 0; i < pCount; ++i)
{
Memory::Memcpy(mData->data() + lItr->mOffset + (pIndex + i)*mVertexSize, pDataValue + (i*lAttributeSize), lAttributeSize);
}
}
}
/*****************************************************************************/
void VertexBuffer::addAttribute(ESemantic pSemantic, EVertexAttributeFormat pFormat)
{
SAttributeDesc lDesc;
lDesc.mSemantic = pSemantic;
lDesc.mFormat = pFormat;
lDesc.mOffset = mVertexSize;
mAttributes.push_back(lDesc);
// Size of one vertex
mVertexSize += TranslateVertexAttributeFormatToByte(pFormat);
}
| code |
ചരമ വാര്ഷികാചരണം നടത്തി വൈക്കം: പി. കൃഷ്ണപിള്ളയുടെ 73ാം ചരമവാര്ഷികവും സിപിഎം വൈക്കം ഏരിയ കമ്മിറ്റി അംഗമായിരുന്ന കെ.വിജയന്റെ 10ാം ചരമ വാര്ഷികവും സിപിഎമ്മിന്റെ നേതൃത്വത്തില് ആചരിച്ചു. വൈക്കം പ്രൈവറ്റ് ബസ് സ്റ്റാന്ഡില് സംഘടിപ്പിച്ച അനുസ്മരണ സമ്മേളനം സിപിഐ ജില്ലാ സെക്രട്ടറിയേറ്റ് അംഗം പി.കെ. ഹരികുമാര് ഉദ്ഘാടനം ചെയ്തു. നവീകരിച്ച സിപിഎം വൈക്കം ടൗണ് നോര്ത്ത് ലോക്കല് കമ്മിറ്റി ഓഫീസിന്റെ ഉദ്ഘാടനവും അദ്ദേഹം നിര്വഹിച്ചു. സിപിഎം ഏരിയ കമ്മിറ്റി അംഗം പി. ശശിധരന് അധ്യക്ഷത വഹിച്ചു. കെ.കെ.ഗണേശന്, കെ. അരുണന്, പി. ഹരിദാസ്, രാഗിണി മോഹന്, എം.സുജിന്, സി.പി. ജയരാജ്, ടി.ജി. ബാബു തുടങ്ങിയവര് പ്രസംഗിച്ചു. | malyali |
#region Copyright & License Information
/*
* Copyright 2007-2011 The OpenRA Developers (see AUTHORS)
* This file is part of OpenRA, which is free software. It is made
* available to you under the terms of the GNU General Public License
* as published by the Free Software Foundation. For more information,
* see COPYING.
*/
#endregion
using System.Collections.Generic;
using OpenRA.Effects;
using OpenRA.Graphics;
using OpenRA.Traits;
namespace OpenRA.Mods.RA.Effects
{
class SatelliteLaunch : IEffect
{
int frame = 0;
Animation doors = new Animation("atek");
WPos pos;
public SatelliteLaunch(Actor a)
{
doors.PlayThen("active",
() => a.World.AddFrameEndTask(w => w.Remove(this)));
pos = a.CenterPosition;
}
public void Tick( World world )
{
doors.Tick();
if (++frame == 19)
world.AddFrameEndTask(w => w.Add(new GpsSatellite(pos)));
}
public IEnumerable<IRenderable> Render(WorldRenderer wr)
{
return doors.Render(pos, wr.Palette("effect"));
}
}
}
| code |
बेबी एबी डी विलियर्स ने अपनी पसंदीदा आईपीएल टीम और क्रिकेटर्स के नाम बताए बेबी एबी डी विलियर्स Baby AB de Villiers के नाम से मशहूर दक्षिण अफ्रीका South Africa के क्रिकेटर डेवाल्ड ब्रेविस Dewald Brevis ने अपनी पसंदीदा आईपीएल फ्रेंचाइजी और क्रिकेटर्स के नाम का खुलासा किया है। बीते दिनों पहले उनकी रॉयल चैलेंजर्स बैंगलोर RCB की पुरानी जर्सी पहने एक तस्वीर सोशल मीडिया पर जमकर वायरल हुई थी। दक्षिण अफ्रीका के अंडर 19 क्रिकेटर ब्रेविस ने कहा है कि वह अपनी पसंदीदा फ्रेंचाइजी आरसीबी के लिए खेलना चाहते हैं, क्योंकि वह टीम उनके दो पसंदीदा खिलाड़ी विराट कोहली Virat Kohli और एबी डी विलियर्स AB de Villiers की है। उन्होंने कहा, मेरे लिए, सबसे बड़ा सपना प्रोटियाज के लिए खेलना है। मैं आईपीएल का बहुत बड़ा फैन हूं और मैं आरसीबी के लिए आईपीएल में खेलना पसंद करूंगा। मैं आरसीबी से प्यार करता हूं, क्योंकि वहां विराट कोहली और एबी डी विलियर्स थे। मैं विराट कोहली और एबी डी विलियर्स का बहुत बड़ा फैन हूं। हरभजन सिंह ने विराट कोहली को चेताया, बोले अब उनपर प्रदर्शन करने का होगा दबाव एबी डी विलियर्स की तरह बेखौफ अंदाज में बल्लेबाजी करने वाले ब्रेविस ने अंडर19 वर्ल्ड कप 2022 में बेहतरीन प्रदर्शन किया है। उन्होंने 4 मुकाबलों में 90.50 के औसत और 86.39 के स्ट्राइक रेट से 362 रन बनाए, जिसमें उच्चतम स्कोर 104 रन का रहा। इस टूर्नामेंट में उन्होंने 1 शतक और 3 अर्धशतक लगाए हैं। | hindi |
राहुल ने इस तरह महात्मा गांधी को किया याद, मच गया बवाल नई दिल्ली। कांग्रेस ने रविवार को महात्मा गांधी की 74वीं पुण्यतिथि पर उन्हें श्रद्धांजलि दी और पार्टी के पूर्व अध्यक्ष राहुल गांधी ने कहा कि हिंदुत्ववादियों को लगता है कि गांधी जी नहीं रहे, लेकिन जहां सत्य है, वहां वह अब भी जीवित हैं। आज ही के दिन 1948 में नाथूराम गोडसे ने महात्मा गांधी की गोली मारकर हत्या कर दी थी। राष्ट्रपिता की पुण्यतिथि को शहीद दिवस के रूप में मनाया जाता है। कांग्रेस नेता ने फॉरएवर गांधी हैशटैग का इस्तेमाल करते हुए ट्वीट किया, एक हिंदुत्ववादी ने गांधी जी को गोली मारी थी। सभी हिंदुत्ववादियों को लगता है कि गांधीजी नहीं रहे। जहां सत्य है, वहां बापू आज भी जीवित हैं। राहुल गांधी ने अपने ट्विटर हैंडल पर महात्मा गांधी की एक टिप्पणी को भी साझा किया, जब मैं निराश होता हूं, मैं याद कर लेता हूं कि समस्त इतिहास के दौरान सत्य और प्रेम के मार्ग पर चलने वाले ही हमेशा विजयी होते हैं। कितने ही तानाशाह और हत्यारे हुए हैं, और कुछ समय के लिए वो अजेय लग सकते हैं, लेकिन अंत में उनका पतन होता हैं। इसके बारे में हमेशा सोचो। भाजपा नेता संबित पात्रा ने इस पर पलटवार करते हुए कहा कि राहुल गांधी की ये निराशावादी राजनीति है। उन्होंने कहा कि नेहरू ने गांधी की सुरक्षा क्यों नहीं बढ़ाई? राहुल गांधी पुराना रिकॉर्ड बजा रहे हैं, उन्हें हिंदुओं का अपमान कर के खुशी मिलती है। कांग्रेस महासचिव प्रियंका गांधी वाड्रा ने भी महात्मा गांधी को श्रद्धाजंलि देते हुए उनका एक वचन ट्वीट किया, अहिंसा कायरता की आड़ नहीं है, बल्कि यह वीर व्यक्ति का सर्वोच्च गुण है। कांग्रेस ने अपने आधिकारिक ट्विटर हैंडल पर कहा, हम राष्ट्रपिता को उनकी पुण्यतिथि पर श्रद्धांजलि देते हैं। इस दिन को शहीद दिवस के रूप में भी मनाया जाता है, हम उन सभी बहादुर पुरुषों और महिलाओं को सलाम करते हैं जिन्होंने देश के लिये अपने प्राणों की आहुति दी। कांग्रेस ने कहा, हमारे प्यारे बापू आज भले ही इस कठिन समय में हमारा नेतृत्व करने के लिए हमारे बीच नहीं हैं, लेकिन अत्याचार, उदासीनता, अन्याय और झूठ के खिलाफ निडर और अथक रूप से लड़ने के उनके तरीके एक समृद्ध और प्रगतिशील भारत की हमारी तलाश में हमारा मार्गदर्शन करते हैं। | hindi |
MES President & CEO Hiten Shah on The Angel Investor’s Network Podcast - MES, Inc.
We are honored to announce MES President & CEO Hiten Shah was a guest on the Angel Investor’s Network podcast with Laura Rubinstein, of Social Buzz Club.
During the interview with Laura Rubinstein, the host of the show, president and CEO of MES, Inc – Hiten talked about his positive spin on failure and how it could always be turned into a lesson on developing better long-term strategies. Hiten also spoke about the importance of defining what you do well, and what you better than anyone else, and how it is critical to redefine that, as clients and the market change continually. Hiten Shah believes that human beings have unlimited potential and he is the living proof.
Having started his career as a plastics engineer, Hiten knew very little about overseeing a global supply chain. But with experience as a key manager, process engineer, engineering manager then director of engineering, vice president of sales, and director of joint venture sales, he was ready for the challenge.
Hiten built MES, Inc. on a culture of trust and honesty, where employees have the freedom to express themselves and where everyone in the company strives to treat all people—coworkers, customers, and suppliers—fairly and transparently. MES today incorporates both his ambitious work ethic and his personal values. | english |
ખાખીજાળીયામાં એન.એસ.એસ. શિબીર યોજાઈ ધોરાજી,તા.24 ત્રિલોક સેવા સમિતિ દ્વારા સંચાલિત ઇવા કોલેજ ઓફ આયુર્વેદ અને ધ્રુવ આયુર્વેદ હોસ્પિટલના ડોક્ટર્સ તથા વિદ્યાર્થીઓ દ્વારા 7 દિવસની એનએસએસ શિબિરનું આયોજન ખાખીજાળીયા ખાતે કરવામાં આવેલ હતું. શિબિર દરમિયાન ગ્રામજનોના સ્વાસ્થ્યની જાગૃતિ અંગે વિવિધ કાર્યક્રમો યોજવામાં આવેલ હતા. આ કાર્યક્રમોમાં ફ્રી નિદાન અને સારવાર કેમ્પ જેમાં 260 દર્દીઓનું ફ્રીમાં તપાસ તથા સારવાર આપવામાં આવી હતી. મહિલા સભા, મહિલા સશક્તિકરણ, ગામની કુમારશાળા તથા ક્ધયાશાળામાં 190 જેટલા બાળકોના આંખ, નાક, કાન, ગળાની તપાસ કરવામાં આવી હતી. શાળાઓમાં વિવિધ સ્વાસ્થ્ય સંબંધી પ્રોગ્રામ કરવામાં આવ્યા હતા. જેમાં બાળકોની ઇમ્યુનિટી વધારવા માટે 270 જેટલા બાળકોને સુવર્ણ પ્રાશનના ટીપા પીવડાવવામાં આવ્યા હતા. તેમજ વિવિધ સાંસ્કૃતિક કાર્યક્રમોનું આયોજન કરવામાં આવ્યું હતું. આ શિબિરમાં ડો. ગુણવંત નાયક, ડો. બલદેવ, ડો. રવિ, ડો. કુણાલ, ડો. ખ્યાતિ, ડો. સ્વાગતા, ડો. પ્રણવ તથા વિદ્યાર્થીઓ દ્વારા સેવા આપવામાં આવી. શિબિરમાં ડો. મનોજ રુંધે પ્રિન્સીપાલ, ડો. નાશીર પરમાર વાઇસ પ્રિન્સીપાલ, ડો. જયશ્રી ગવાંડે, તાલુકા પંચાયત પ્રમુખ વિનુભાઈ ચંદ્રવાડીયા, તાલુકા પંચાયત સભ્ય ભરતભાઈ, માજી તાલુકા પંચાયત પ્રમુખ લાખાભાઈ, ભાવેશભાઈ સરપંચ ખાખીજાળીયા, રાજુભાઈ ગર સરપંચ મોજીરા, વિક્રમભાઈ સરપંચ કેરળા, પોલાભાઈ સરપંચ સેવન્ત્રા, તથા આજુબાજુના ગામના આગેવાનોની તથા ઇવા કોલેજ ઓફ આયુર્વેદના સ્ટાફની વિશેષ ઉપસ્થિત રહી હતી. | gujurati |
సాయి పల్లవి ఓటీటీ ఆశలు విరాటపర్వం సినిమాపై హీరోయిన్ గంపెడు ఆశలు పెట్టుకుంది. కానీ అవి నెరవేరలేదు. వేణు ఉడుగుల దర్శకత్వంలో రానా దగ్గుబాటిసాయి పల్లవి జంటగా విరాటపర్వం తెరకెక్కింది. రిలీజ్ కు ముందు ఈ సినిమాపై పాజిటివ్ బజ్ ఏర్పడింది. ఇప్పటికే సినిమా చూసేశాం. అద్భుతంగా వచ్చింది. వెన్నెల పాత్రలో సాయి పల్లవి ఇరగదీసింది. ఆమెకు జాతీయ అవార్డు ఖాయమని విక్టరీ వెంకటేష్ లాంటోళ్లు బల్లగుద్ది మరీ చెప్పారు. కానీ జూన్ 1న ప్రేక్షకుల ముందుకొచ్చిన విరాటపర్వం వెంకీ చెప్పిన రేంజ్ లో లేదని ప్రేక్షకులు తీర్పునిచ్చారు. నిజంగా వెన్నెల పాత్రను దర్శకుడు అద్భుతంగా ప్రజెంట్ చేశారు. అయితే కథ బాగున్నా.. కథనం గ్రిప్పింగ్ గా లేకపోవడంతో రిజల్ట్ తేడా కొట్టేసింది. దీంతో విరాటపర్వం మంచి సినిమా అనిపించుకున్నా.. కమర్షియల్ గా పెద్దగా ఆడలేదు. హిట్ సినిమా అనిపించుకోలేదు. అయితే థియేటర్స్ లో ఆడని కొన్ని సినిమాలు ఓటీటీల్లో అదరగొడుతున్నాయి. జులై 1న విరాటపర్వం ఓటీట్రీలో రిలీజ్ కానుంది. నెట్ ఫ్లిక్స్ లో స్ట్రీమింగ్ కానుంది. ఈ నేపథ్యంలో విరాటపర్వం ఓటీటీలో బాగా ఆడుతుందని సాయి పల్లవి ఆశలు పెట్టుకున్నట్టు కనిపిస్తుంది. | telegu |
Varudu Kavalenu: ఘనంగా నిర్వహించిన వరుడు కావలెను సక్సెస్ మీట్. Varudu Kavalenu: యువ హీరో నాగశౌర్య, పెళ్లి చూపులు బ్యూటీ రీతూ వర్మ జంటగా నటించిన లేటెస్ట్ ఫ్యామిలీ డ్రామా వరుడు కావలెను. ఈ సినిమా అక్టోబర్ 29 న విడుదల ప్రపంచవ్యాప్తంగా విడుదలై మంచి ఆదరణ పొందుతోంది. దీంతో చిత్ర బృందం తాజాగా సక్సెస్ మీట్ ఏర్పాటు చేసింది. దీనికి సంబంధించిన పిక్స్ ప్రస్తుతం వైరల్గా మారాయి. ఈ సినిమాను లక్ష్మి సౌజన్య దర్శకత్వం వహించగా సితార ఎంటర్టైన్మెంట్స్ బ్యానర్పై సూర్యదేవర నాగవంశీ నిర్మించారు. ఈ చిత్రానికి సంగీతం విశాల్ చంద్ర శేఖర్ అందించారు. ఇక ఆ మధ్య ఈ సినిమా నుంచి ఒక మాస్ మసాలా సాంగ్ను చిత్రబృందం విడదల చేసిన సంగతి తెలిసిందే. దిగు దిగు దిగు నాగ అంటూ సాగే ఈ పాట ఇన్స్స్టాంట్ రెస్పాన్స్ను దక్కించుకుంది. తెలంగాణ జానపదం దిగు దిగు దిగు నాగ అనే పాటను మార్చి అదే బాణీలో కొత్త లిరిక్స్తో అదరగొట్టారు. ఈ పాటకు అనంత్ శ్రీరామ్ సాహిత్యాన్ని సమకూర్చారు. ప్రముఖ హిందీ సింగర్ శ్రేయా ఘోషల్ ఈ పాటను పాడారు. యూట్యూబ్లో విడుదలై ఈ పాట నెటిజన్స్ను ఎంతోగాను ఆకట్టుకుంటోంది. ఈ సినిమా రెండు తెలుగు రాష్ట్రాల్లో మొదటిరోజు బాక్సాఫీస్ దగ్గర ఒక కోటి నుండి 1.2 కోట్ల రేంజ్లో కలెక్షన్స్ ని అందుకునే ఛాన్స్ ఉందని భావించినా.. 96 లక్షల రేంజ్ షేర్తో పరవాలేదనిపించింది. వరుడు కావలెను సినిమాను మొత్తంగా 8.6 కోట్ల రేంజ్ రేటుకి వరల్డ్ వైడ్గా అమ్మారు. దీంతో ఈ సినిమా బాక్స్ ఆఫీస్ దగ్గర 9 కోట్లు టార్గెట్తో బరిలోకి దిగింది. దీంతో ఫస్ట్ డే సాధించిన కలెక్షన్స్ కాకుండా వరుడు కావలెను సినిమా ఇంకా 7.64 కోట్ల రేంజ్లో షేర్ను అందుకోవాల్సి ఉంటుంది. | telegu |
Vitamin D: ভিটামিন ডির ঘাটতি? চটজলদি বুঝে নিন স্বাদ ও গন্ধের পরীক্ষায় ভিটামিন ডি শরীরের জন্য একটি অত্যন্ত গুরুত্বপূর্ণ উপাদান এটি ক্যালসিয়াম শোষণ করতে শরীরকে সাহায্য করে পাশাপাশি বজায় রাখে হাড় ও পেশীর স্বাস্থ্য কিন্তু খাবারদাবার থেকে সচরাচর এই উপাদান সহজে পাওয়া যায় না তাই অনেক ক্ষেত্রেই ভিটামিন ডির ঘাটতি দেখা যায় শরীরে সাধারণত মানুষের ত্বক সূর্যালোকের সংস্পর্শে এলে দেহ থেকেই এই ভিটামিন উত্পন্ন হয় এই ভিটামিন কম থাকলে নানা ধরনের সমস্যা দেখা যেতে পারে আর তেমনই একটি উপসর্গ হল স্বাদ ও গন্ধের অনুভূতি হ্রাস পাওয়া একটি বিজ্ঞান বিষয়ক পত্রিকায় প্রকাশিত গবেষণাপত্র বলছে, বয়সের সঙ্গে সঙ্গে স্বাদ ও গন্ধের অনুভূতি হারিয়ে ফেলার যোগসুত্র রয়েছে শরীরে এই ভিটামিনের অভাবের সঙ্গে বিশেষজ্ঞদের মতে, মানুষ মোট আটটি গন্ধ পায় তার মধ্যে ছয়টির বেশি গন্ধ চিহ্নিত করতে না পারলে তাকে গন্ধের অনুভূতির হ্রাস পাওয়ার লক্ষণ এবং খাদ্য লবণের স্বাদ অনুভব না করতে পারলে তাকে স্বাদের অনুভূতি হ্রাস পাওয়ার লক্ষণ হিসেবে চিহ্নিত করা যায় গবেষণা বলছে যাঁদের শরীরে পর্যাপ্ত পরিমাণে ভিটামিন ডি উত্পন্ন হয় না বা যাঁদের ভিটামিনের ঘাটতি রয়েছে, বেশি বয়সে তাঁদের এই রকম স্বাদ ও গন্ধ হারিয়ে ফেলার আশঙ্কা একজন সুস্থ ব্যক্তির তুলনায় ৩৯ শতাংশ বেশি কাজেই বিজ্ঞানীরা কার্যত নিশ্চিত যে, ভিটামিন ডির সঙ্গে স্বাদ ও গন্ধ লোপ পাওয়ার কোনও না কোনও যোগাযোগ রয়েছে ফলে এ বিষয়ে সজাগ থাকুন প্রয়োজনে পরামর্শ নিন চিকিত্সকের | bengali |
\begin{document}
\title{Simulating quench dynamics on a digital quantum computer with data-driven error mitigation}
\author{Alejandro Sopena}
\author{Max Hunter Gordon}
\author{Germ\'{a}n Sierra}
\author{Esperanza L\'{o}pez}
\affiliation{Instituto de F\'{\i}sica Te\'{o}rica, UAM/CSIC, Universidad Aut\'{o}noma de Madrid, Madrid, Spain}
\begin{abstract}
Error mitigation is likely to be key in obtaining near term quantum advantage. In this work we present one of the first implementations of several Clifford data regression based methods which are used to mitigate the effect of noise in real quantum data. We explore the dynamics of the 1-D Ising model with transverse and longitudinal magnetic fields, highlighting signatures of confinement. We find in general Clifford data regression based techniques are advantageous in comparison with zero-noise extrapolation and obtain quantitative agreement with exact results for systems of $9$ qubits with circuit depths of up to $176$, involving hundreds of CNOT gates. This is the largest systems investigated so far in a study of this type. We also investigate the two-point correlation function and find the effect of noise on this more complicated observable can be mitigated using Clifford quantum circuit data highlighting the utility of these methods.
\end{abstract}
\maketitle
\section{Introduction}
The rapid progress in the field of quantum computing is encouraging, with current machines approaching the qubit qualities and system sizes expected to demonstrate some useful quantum advantage. However, noise within the computation still presents a large obstacle in obtaining useful results as current systems cannot implement full error correction. Therefore, it is expected that error mitigation techniques will be essential in demonstrating useful quantum advantage. These techniques aim to reduce the impact of noise rather than remove its effects completely. This relatively new field is experiencing a period of rapid progress with novel methods being developed in quick succession. Common approaches include quantum circuit compiling, machine learning~\cite{Murali,Cincio_2018, cincio2020machine} and variational algorithms~\cite{Peruzzo2014, cerezo2020variationalreview, sharma2019noise, O'Malley2016, cirstoiu2020variational}. Recent advances show phase estimation~\cite{o2020error} and so-called virtual state distillation~\cite{koczor2020exponential, huggins2020virtual, czarnik2021qubitefficient} can also be used for error mitigation and show great promise.
One of the most popular techniques is so called zero-noise extrapolation (ZNE)~\cite{Temme_2017}. Data from an observable of interest evaluated at several controlled noise levels is used to give an improved estimate of the noise free observable. Despite much success~\cite{otten2019recovering, Dumitrescu_2018, Kandala_2019, cai2020multiexponential} this technique is limited by the assumption of low hardware noise, which may not be valid in the circuits of a size and depth necessary to demonstrate quantum advantage.
Recently, it has been shown that data sets produced by classically simulable quantum circuits such as near-Clifford circuits~\cite{czarnik2020error, strikis2020learning}, circuits based on fermionic linear optics or matchgate circuits~\cite{montanaro2021error} can be used to mitigate the effects of noise. In so called Clifford data regression (CDR) the exact and noisy data from near-Clifford circuits is used to learn a functional relation between the noisy and exact observables. This relation can then be applied to a noisy observable of interest which cannot be simulated classically. This technique can also be unified with ZNE~\cite{lowe2020unified}. In variable noise Clifford data regression (vnCDR) near-Clifford circuits are evaluated at several controlled noise levels. The exact and noisy values are then used to perform a guided extrapolation to the zero-noise limit. In general, these regression based methods appear advantageous over ZNE due to their simplicity and scalability. However, there are few examples of these methods being applied to real data from currently available quantum computers, where noise is significantly more difficult to mitigate.
One clear application of quantum technologies is the simulation of quantum systems. The classical resources necessary to simulate such systems in general scales exponentially with the system size. Spin - $\frac{1}{2}$ systems are particularly relevant as they map directly onto physical qubits, making spin chains an ideal testing ground for both current and future quantum computers~\cite{Smith_2019}.
A common problem to consider in condensed matter simulations is non-equilibrium dynamics. These dynamics can be induced by a global quench, which is a sudden change to the system Hamiltonian.
Simulations obtaining quantitative accuracy have been reported in ion trap architectures~\cite{tan2019observation} and more recently in super-conducting architectures~\cite{vovrosh2021efficient}.
In this work we provide a comparison of several error mitigation strategies applied to the problem of simulating a quantum quench in the one dimensional Ising model with transverse and longitudinal magnetic fields. To investigate how these methods perform in real quantum devices we explore the behavior of several observables of interest and simulate the system dynamics with various circuit depths. We measure the frequency of oscillations of the magnetisation for different initial states in a system of $9$ spins. Furthermore, we present one of the first measurements of the two-site correlation function in a study of these dynamics on a superconducting device. We are able to mitigate the effect of noise in data produced by deeper circuits and larger systems than previously explored in similar works. We find that Clifford regression based methods are able to obtain quantitative accuracy with the exact results and consistently outperform ZNE.
First, we present an overview of the techniques used in data-driven error mitigation, with particular focus on CDR and vnCDR and their relation to recent advances. A slight variation of these methods is explored, namely "poor-man's" CDR (pmCDR), which performs well for short depth trotterised simulations of hamiltonian dynamics. We then review the theoretical expectations of the model and the methods used to implement the simulation in a super-conducting architecture. We show that using error mitigation we are able to obtain consistent quantitative accuracy involving circuits of depth $110$ with $160$ CNOT gates and $9$ qubits, while also obtaining some results with quantitative accuracy for depths up to $176$ with $256$ CNOT gates. Finally we conclude with a discussion of the results presented here and an outline of future directions.
\section{Data-driven error mitigation}
Data-driven error mitigation uses classical post processing of quantum data to improve the zero-noise estimates of some observable of interest. In this work ZNE~\cite{Temme_2017}, CDR~\cite{czarnik2020error} and vnCDR~\cite{lowe2020unified} are used to obtain noise-free estimates of various observables. Furthermore, following the recent work showing the success of a simple mitigation strategy with an assumed noise model~\cite{vovrosh2021efficient}, we demonstrate the utility of a similar approach where the parameters of an assumed noise model are learned using near-Clifford circuits (pmCDR).
\subsection{ZNE}
Zero-noise extrapolation is one of the most popular error mitigation strategies. It uses quantum circuit data collected at various hardware noise levels to estimate the value of a noise free observable. Intuitively, by increasing the noise in a controlled manner and extrapolating to the zero-noise limit one can obtain a more accurate estimate of an observable of interest. Originally, this technique was presented within the context of stretching gate times to increase noise and using Richardson extrapolation to approach the zero-noise limit~\cite{Temme_2017}. More recently this has been extended to hardware agnostic approaches through unitary folding~\cite{giurgicatiron2020digital, larose2020mitiq} and identity insertion methods~\cite{He_2020}. Furthermore, additional extrapolation techniques have been proposed~\cite{PhysRevX.8.031027, otten2019recovering}.
Despite widespread success ZNE performance guarantees are limited due to uncertainty in the extrapolation. Additionally, in real devices often the base-level noise is too strong to enable an accurate extrapolation, particularly in circuits with significant depth.
\subsection{CDR}
More recently, Clifford circuit quantum data has been used to mitigate the effect of noise~\cite{czarnik2020error, strikis2020learning}. Quantum circuits composed of mainly Clifford gates can be evaluated efficiently on a classical computer. In CDR near-Clifford circuits are used to construct a set of noisy and exact expectation values for some observable of interest. This dataset is used to train a simple linear ansatz mapping noisy to exact values. Following the presentation in~\cite{czarnik2020error} taking $\hat{\mu}_{0}$ to be the observable evaluated with hardware noise, CDR uses Clifford circuits to train the following anstaz:
\begin{equation}
f(\hat{\mu}_0) = a_1\hat{\mu}_0 + a_2
\label{eq:cdr_ansatz}
\end{equation}
The parameters $a_1$, $a_2$ are chosen using least-squares regression on the near-Clifford circuit dataset. For a training set of $m$ Clifford circuits with noisy expectation values $\{x_i\}$ and exact expectation values $\{y_i\}$ evaluated classically, one calculates
\begin{equation}
(a_1^{*},a_2^{*}) = \underset{(a_1,a_2)}{\text{argmin}} \sum_{i=1}^m \left[y_i - (a_1 x_i + a_2)\right]^2.
\end{equation}
These learned parameters are then used to mitigate the effect of noise on an observable produced by a circuit which is not classically simulable. As noted in Ref.~\cite{czarnik2020error} the form of the anstaz can be motivated by considering the action of a global depolarizing channel. Letting $\rho$ be the density matrix for the noise-free state after some evolution.
Consider the action of a depolarizing noise channel $\mathcal{E}$ which acts on this state before a measurement of some observable of interest $X$. The action of the channel can be described as follows
\begin{align}
\tr(\mathcal{E}(\rho) X) = (1-\epsilon)\tr(\rho X) + \frac{\epsilon\tr(X)}{d}
\label{eq:depolarising}
\end{align}
where $d$ is the dimension of the system and $\epsilon\in\qty(0,1)$ is a parameter characterizing the noise. Identifying $\hat{\mu}_0 = \tr(\mathcal{E}(\rho) X)$ and
\begin{align}
a_1 = 1/(1-\epsilon),\quad a_2 = -\frac{\epsilon}{d(1-\epsilon)}\tr(X)
\label{eq:clifford_parameters}
\end{align}
the noise-free expectation value $\tr(\rho X)$ can be calculated using Eq.~\eqref{eq:cdr_ansatz}. Therefore, in the case of a global depolarizing channel CDR should perfectly mitigate the noise, assuming the Clifford circuit training set accurately captures its effect. This ansatz also perfectly corrects certain types of measurement error~\cite{czarnik2020error}.
For observables $X$ with $\tr(X) = 0$ the linear term in CDR appears to be redundant assuming the noise can be modelled with a global depolarising channel, shown to be an accurate description in some circumstances~\cite{vovrosh2021efficient}. However, we find including the constant term in the anstaz allows for more flexible fitting of the training data, leading to a better mitigation in general (\textit{e.g.} for the data shown in Fig.~\ref{fig:magnetisation} the absolute error is improved by a factor of $1.2$). An example of such a case can be seen in Appendix~\ref{sec:CDR_cnst}.
\subsection{Poor man's CDR}
\label{sec:pmcdr}
As previously mentioned, recently it has been shown that a global depolarising channel (Eq.~\ref{eq:depolarising}) appears to accurately describe the noise in a real device for small system sizes~\cite{vovrosh2021efficient, urbanek2021}. Indeed, this noise model provides the motivation for use of a linear anstaz in CDR. Here, we implement a simplified version of CDR where short depth near-Clifford circuits are used to fit the parameter $\epsilon$ characterising a global depolarising noise channel:
\begin{equation}
\expval{X}_{noisy} = (1-\epsilon)\expval{X}_{exact} +\frac{\epsilon \tr(X)}{d}.
\label{pmcdr_ansatz}
\end{equation}
Training sets are constructed from the quantum circuits of one and two Trotter steps. This data is used to determine $\epsilon_1$ and $\epsilon_{2}$ respectively. Due to the repetitive structure of the circuit in a trotterised evolution we assume the effect of the error on an observable can be modelled using Eq.~\eqref{pmcdr_ansatz}, with the parameters evolving as $(1-\epsilon)^{\alpha N_{T}}$, where $N_{T}$ is the number of Trotter steps and $\alpha$ is some constant (see also~\cite{vovrosh2021efficient}). Using $\epsilon_{1}$ and $\epsilon_{2}$, determined with the near-Clifford training data, we can fit $\alpha$ and use this assumed model to correct observables from circuits involving more Trotter steps. We find $\alpha$ is close to one for the magnetisation while for $\Delta_i^{ZZ}(t)$ it is higher (\textit{e.g.} the mean values of $\alpha$ are 1.01 and 1.30 for the magnetisation and $\Delta_i^{ZZ}(t)$ results shown in Figs.~\ref{fig:magnetisation} and.~\ref{fig:delta} respectively).
The advantage of this approach is that it is only necessary to produce two near-Clifford data sets. The parameters of the noise model are then learned and applied to observables from other circuits. This is more convenient, having a reduced experimental and computational overhead. We note a similar technique was recently presented using estimation circuits consisting of only CNOT gates and combining this idea with randomised compiling and zero-noise extrapolation~\cite{urbanek2021}.
\subsection{vnCDR}
Zero noise extrapolation and Clifford data regression can be conceptually unified into one mitigation strategy where Clifford circuit quantum data is used to inform the functional form of the extrapolation to the zero-noise limit~\cite{lowe2020unified}. Intuitively, so called variable noise Clifford data regression reduces the risk of blind extrapolation and is expected to outperform both ZNE and CDR in deep quantum circuits involving many qubits. vnCDR makes use of Clifford circuits evaluated at several noise levels to train a more general anstaz than that of CDR. Considering $m$ near-Clifford circuits and $n+1$ noise levels $c_j\in \mathcal{C}$, a noisy estimate of the observable expectation value is defined as $x_{i, j}$. For each of the $m$ circuits the corresponding exact observable $y_i$ is computed classically. The training set $\mathcal{T}$ is taken as $\mathcal{T} = \{(\vec{x}_i, y_i)\}$ where $\vec{x}_i = (x_{i,0},\dots,x_{i,n})$ is the vector of noisy expectation values produced by the $i^\text{th}$ circuit. This training data is used to learn a function that takes a set of noisy estimates at the $n+1$ different noise levels and outputs an estimate for the noise-free value. We use the linear ansatz
\begin{equation}
g(\vec{x}; \vec{a}, b) = \vec{a}\cdot\vec{x} + b \,,
\end{equation}
where we have included a constant term $b$. Least-squares regression is used on the dataset $\mathcal{T}$ to pick optimal parameters $\vec{a}^*, b^*$, i.e.,
\begin{align}
(\vec{a}^*, b^*) = \underset{\vec{a}, b}{\text{argmin}} \sum_{i=1}^m \left(y_i - (\vec{a}\cdot \vec{x}+ b)\right)^2\,.
\end{align}
Therefore, $g(\vec{x}; \vec{a}^*, b^*)$ is expected to output a good estimate for the noise-free expected value from a vector formed of the noisy expectation values at different controlled noise levels. This mitigation strategy is also expected to perfectly mitigate for a global depolarising noise channel. Despite promising results the performance of vnCDR has not been extensively explored on real quantum circuit data, motivating the analysis we present here.
Originally vnCDR was introduced with an ansatz excluding the constant term $b$, appearing more similar to Richardson extrapolation~\cite{lowe2020unified}. For the observables we consider here including this parameter made for a more accurate mitigation (\textit{e.g.} for the data shown in Fig.~\ref{fig:magnetisation} the absolute error is improved by a factor of $1.1$).
Overall, classically simulable near-Clifford circuits can be used to inform the experimenter about the noise present in the device. CDR makes use of extracting this data for every circuit to mitigate the results of an observable of interest from that particular circuit. Assuming a noise model, pmCDR makes use of two Clifford data sets and uses this data to complete a mitigation on circuits with a repetitive structure at different depths. Data collected at various artificial controlled noise rates also contains relevant information to perform a mitigation as shown in ZNE. vnCDR conceptually unifies ZNE and CDR by collecting near-Clifford circuit data at various noise levels.
\section{Model}
Data-driven error mitigation is a promising approach to reduce the impact of noise in near term quantum computers. One of the areas where quantum algorithms are expected to show some advantage over classical methods is in simulating quantum many body systems. A system which displays interesting many body dynamics is the TFIM with an additional longitudinal field, providing a clear test bed for these mitigation methods.
\subsection{Transverse-Longitudinal Ising model}
The Hamiltonian of the quantum one-dimensional Ising model of length $L$ with transverse and longitudinal fields is given by
\begin{equation}\label{IisingHamiltonian}
H=-J\left[\sum_{i}^{L} \hat{\sigma}_{i}^{Z} \hat{\sigma}_{i+1}^{Z}+h_{X} \sum_{i=1}^{L} \hat{\sigma}_{i}^{X}+h_{Z} \sum_{i=1}^{L} \hat{\sigma}_{i}^{Z}\right]
\end{equation}
where $J$ is an exchange coupling constant, which sets the microscopic energy scale and $h_X$ and $h_Z$ are the transverse and longitudinal relative field strengths, respectively. This model is integrable for $h_Z=0$ while for $h_Z\neq0$ it is only integrable in the continuum when $h_X=1$~\cite{zamolodchikov_integrals_1989}.
Setting $h_Z=0$, in the continuum limit, the diagonalisation of the Hamiltonian results in the description of a fermion with mass $m=2J\abs{1-h_X}$ and velocity $v=2J\sqrt{h_X}a$ where $a$ is the chain spacing and $ka\ll1$~\cite{sachdev_quantum_2011}. At $h_X=1$, the system has a critical point and the low-energy behaviour of the system is described by a conformal field theory with central charge $c=1/2$~\cite{mussardo_statistical_2010}. For $h_X<1$, the system is in the ferromagnetic phase (with $J>0$). This system can be approximated by considering the low-energy elementary particle excitations which are given by domain walls between the two ground states of $H$ with $h_X=0$~\cite{sachdev_quantum_2011},
\begin{equation}
\ket{i}=\ket{\uparrow\cdots\uparrow_{i-1}\uparrow_{i}\downarrow_{i+1}\downarrow_{i+2}\downarrow_{i+3} \cdots\downarrow}.
\end{equation}
These states are identified as fermions.
A longitudinal field $h_Z\neq0$ induces a confining potential between pairs of domain walls,
\begin{equation}
\ket{i,n}=\ket{\uparrow\ldots\uparrow_{i-1}\downarrow_{i} \ldots \downarrow_{i+n-1} \uparrow_{i+n} \ldots \uparrow},
\label{eq:meson}
\end{equation}
which increases linearly with the length of the domain, $n$. This leads to excitations formed from pairs of domain walls Eq.~\eqref{eq:meson}, which are referred to as mesons~\cite{McCoy:1994zi}.
In order to show the temporal evolution of the position of fermions and mesons, we measure the probability distribution of kinks,
\begin{equation}
\Delta_i^{ZZ}(t)=\frac{1}{2}(1-\expval{\hat{\sigma}_i^Z\hat{\sigma}_{i+1}^Z}),
\end{equation}
from an initial state of $2$ kinks (see Appendix~\ref{sec:masses}). This observable takes the value $0$ when there are no kinks and is $1$ when the $i$-th and $(i+1)$-th spins form a kink.
Confinement suppresses the light cone spreading of correlations~\cite{kormos_real-time_2017}. This effect can be seen by measuring the two point correlation function,
\begin{equation}
\sigma_{i, j}^{Z Z}(t)=\expval{\hat{\sigma}_i^Z\hat{\sigma}_j^Z}-\expval{\hat{\sigma}^Z_{i}}\expval{\hat{\sigma}_j^Z}.
\end{equation}
In the presence of a longitudinal field and $h_X<1$ local observables after quenches exhibit oscillations whose dominant frequencies are the energy gaps between bound states~\cite{kormos_real-time_2017} (see Appendix~\ref{sec:masses}). These energy gaps can be interpreted as meson masses. A suitable observable for measuring the meson masses is the magnetisation, $\sigma_i^Z(t)=\expval{\hat{\sigma}_i^Z}$. In order to avoid edge effects we measure the magnetisation at the centre of the chain for initial states without domain walls and at the outer edge of the domain wall for initial states of two domain walls~\cite{tan2019observation}.
In this work we explore the signatures of confinement by measuring the probability distribution of kinks $\Delta_i^{ZZ}(t)$, the evolution of the two point correlation function $\sigma_{i, j}^{Z Z}(t)$ and the meson masses determined by extracting the dominant frequency of the oscillation of the magnetisation $\sigma_{i}^Z$.
\subsection{Quantum simulation}
We simulate the induced Hamiltonian dynamics using a first order trotterised evolution of the initial state. We start by discretising the evolution operator in $n$ blocks such that
\begin{equation}
U(t)=e^{-iHt}=\qty(e^{-iH\Delta t})^n=U^n\qty(\Delta t)
\end{equation}
with $\Delta t=t/n$. Each evolution step operator is approximated using the first order Trotter expansion:
\begin{equation}
U(\Delta t)=e^{-iH\Delta t}=\prod_{k} e^{-i h_{k} t / n}+\mathcal{O}((\Delta t)^{2})
\end{equation}
where $h_k=-J\left[ \hat{\sigma}_{k}^{Z} \hat{\sigma}_{k+1}^{Z}+h_{X} \hat{\sigma}_{k}^{X}+h_{Z} \hat{\sigma}_{k}^{Z}\right]$. To implement $e^{-i h_{k} t / n}$ on IBM devices, we decompose the quantum circuit to execute one Trotter step into the native IBM gate set $\{R_{X}(\pi/2), R_{Z}(\theta), X, \text{CNOT}\}$ (see Appendix~\ref{sec:circuit}). This decomposition leads to a depth of $11$ per Trotter step with $2(Q-1)$ CNOT gates for a system size $Q>2$, where $Q$ is the number of qubits. For a fixed time step $\delta t$ one can evaluate the dynamics up to time $t$ by repeated action of this circuit $N_{T}$ times, where $N_{T}=t/\delta t$.
\begin{figure*}
\caption{Temporal evolution of $Z$-axis local magnetisation with $h_X=0.5$ and $h_Z=0.9$ for three initial states (in each row) with the observables mitigated using various techniques (columns). In all panels the exact diagonalised dynamics is shown as a black-solid line. Raw observables are shown in the left most column ($a$) calculated at two noise levels $\mathcal{C}
\label{fig:magnetisation}
\end{figure*}
\section{Simulated Spin chain confinement}
In this section we display the results obtained after applying the mitigation methods described above on the trotterised evolution of a system of $Q=9$ qubits. We investigate three observables of interest: the magnetisation, $\sigma^{Z}_i$, to determine the masses of the mesons, $\Delta^{ZZ}_i$, to visually demonstrate confinement and two-point correlation function, $\sigma^{ZZ}_{ij}$, to explore how the mitigation techniques perform on a more complex, non-local observable. Every circuit used, both in training set construction for mitigation and in collecting raw data, was evaluated with $8192$ shots. We use the absolute error to quantitatively explore the performance of the mitigation strategies implemented:
\begin{equation}
error = \left|\frac{\langle X \rangle_{mitigated} - \langle X \rangle_{trotterised}}{mean(\langle X \rangle_{trotterised})}\right|,
\end{equation}
where the mean is taken over the time evolution.
\subsection{Local magnetisation evolution}
To determine the first meson masses we measure the oscillations of the local magnetisation for three initial states. We extract the dominant frequencies using a single-frequency sinusoidal fit as in Ref.~\cite{vovrosh2021efficient},
\begin{equation}
\sigma^Z_j=a_1e^{-a_2t}\cos(a_3t)+a_4t+a_5.
\label{eq:fit}
\end{equation}
We explore the evolution starting with the system initialised as: all spins up, the central qubit in the down state and all other qubits up and two central qubits in the down state with all others spin up. The system is evolved using a trotterised evolution of the Hamiltonian for a fixed time step. The circuit depth therefore grows linearly with the number of Trotter steps. When the system is initialised to all spins up, the Trotter step was chosen to be $0.25J$ leading to a final circuit depth of $176$ with $256$ CNOT gates. In both other initialisations the Trotter step was $0.5J$ leading to a final circuit depth of $110$ involving $160$ CNOT gates. A different Trotter step is needed in these two cases to reproduce the smaller amplitude and higher frequency oscillations observed with the initial state being all spins up.
The temporal evolution of the magnetisation is shown in Fig.~\ref{fig:magnetisation}. The raw values for the magnetisation clearly decay towards the maximally mixed state with circuit depth. With a higher noise level this can be seen to occur more quickly, as expected. We mitigate the raw results using ZNE, CDR, vnCDR and pmCDR.
From the time evolution of the magnetisation for different values of $h_z$ (see Appendix~\ref{sec:data}) we obtain the dominant frequencies shown in Fig.~\ref{fig:freq_0.5}. In order to calculate the frequencies it is not necessary to fit the entire evolution of the magnetisation. We found more accurate values are obtained by fitting times up to around $tJ=3$.
\begin{figure}
\caption{Frequencies obtained at $h_X=0.5$ and various $h_{Z}
\label{fig:freq_0.5}
\end{figure}
\begin{figure*}
\caption{The observable $\Delta_{i}
\label{fig:delta}
\end{figure*}
All mitigation methods improve upon the raw data. In particular CDR, vnCDR and pmCDR mitigate the effect of noise effectively in many cases, even for deep circuits. In some cases, like those shown in Fig.~\ref{fig:freq_0.5}, pmCDR performs as well as the other Clifford based methods. This suggests due to the repetitive structure of the quantum circuit the effect of noise on this observable can be characterised more easily, using only the first two Trotter steps. However, we find this is not as reliable as using a training set to learn the noise at each Trotter step, as done in CDR and vnCDR. This indicates the noise parameters can vary considerably beyond some circuit depth and also in short time frames, between runs. Still, it is quite remarkable that a simple global noise model describes the noise so accurately in some runs (see Appendix~\ref{sec:data} for more examples).
For more complicated observables treating the noise as purely depolarising breaks down more quickly and pmCDR begins to perform worse. It should be noted that we do not implement measurement error mitigation. While we do not expect this to impact the performance of CDR or vnCDR it should lead to worse implementations of both pmCDR and ZNE. We also do not enforce any physical constraints on our mitigated observables, in order to asses the raw potential of the method. Therefore, occasionally pmCDR gives unphysical values, increasing the absolute error of the mitigation significantly.
Overall, CDR and vnCDR show the most reliable mitigation of the magnetisation. They consistently offer a quantitatively accurate mitigation for times up to $tJ\sim5$ and occasionally up to even longer times. Therefore, it can be concluded the computational overhead necessary for these methods is useful in mitigating the effect of noise. vnCDR does not offer any significant visual advantage over CDR although it does lead to the most accurate calculations of the frequencies and often has a smaller absolute error on average. ZNE and pmCDR perform consistently well for shorter depth circuits.
\subsection{Kink evolution}
Starting with the initial state $\ket{\uparrow\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\uparrow\uparrow}$, projecting into the two kink subspace and measuring the observable $\Delta_{i}^{ZZ}$ shows the evolution of the kinks with time. Confinement can be directly observed by comparing the evolution when $h_{Z} = 0$ and $h_{Z} \neq 0$. With the presence of some transverse field we observe an attraction between the kinks due to the confining potential. These dynamics are more difficult to mitigate in comparison with the local magnetisation as to observe this effect the entire state of the system is probed. This gives a good indication of average performance of each method. In Fig.~\ref{fig:delta} we show the evolution of $\Delta_{i}^{ZZ}$ with the various mitigation methods for $h_{X} = 0.5$ and $h_{Z} = 0,\text{ }0.5,\text{ }0.9$. We note that when $h_{Z}=0$ there are $9$ less non-Clifford gates per Trotter step.
The oscillations that appear to be washed out in the raw data are clearly recovered by CDR and vnCDR. ZNE does partially recover the oscillations but not to the same accuracy. It is important to note that pmCDR begins to fail at around $tJ = 3$ leading to an increased absolute error in comparison with the raw results.
In the pmCDR implementation since $\tr\qty(\Delta^{ZZ})\neq0$, the ansatz relating the mitigated and the noisy observables Eq.~\eqref{pmcdr_ansatz} has a linear term and a constant term dependent on the parameter $\epsilon$. Therefore, to obtain an error of the same magnitude as that of the magnetisation, the difference between the true $\epsilon$ parameter and the one obtained from the fit must be smaller for $\Delta_{i}^{ZZ}$ than for the magnetisation. If during operation the true noise model changes slightly this has a large impact on the results. In addition, because $\Delta_{i}^{ZZ}$ is a two-qubit observable, the ansatz is perhaps too simple to fully characterise the noise. The impact of measurement error is also detrimental. Furthermore, we note the results from pmCDR could be improved by enforcing physical constraints on the mitigated values.
Without mitigation the dynamics of the observables is not significantly changed by the introduction of a transverse field. The separation velocity does appear to be reduced but no oscillatory dynamics is observed. However, with CDR and vnCDR these oscillations are recovered and there is a striking visual contrast between the dynamics with and without the presence of a longitudinal field. We deduce from these results that CDR and vnCDR appear to be the more powerful mitigation strategies.
\subsection{Correlation evolution}
We also investigate the correlation with the central qubit as the system evolves. This observable is non-local and is formed by combining three observables $\expval{\hat{\sigma}_i^Z\hat{\sigma}_5^Z}$, $\expval{\hat{\sigma}^Z_{i}}$ and $\expval{\hat{\sigma}^Z_{5}}$, which we mitigate separately before combining. In general, the correlation decreases with $h_Z$ while $\expval{\hat{\sigma}_i^Z\hat{\sigma}_5^Z}$ and $\expval{\hat{\sigma}^Z_{5}}$ increase. As the longitudinal field increases, it becomes more complicated to mitigate the correlation since the difference between the values of the two correlation terms needs to be smaller. Therefore, the mitigation needs to perform very effectively on each term and the correlation proves challenging to mitigate for general values of $h_{Z}$ and $h_{X}$. Thus, we focus on the case with $h_Z < h_X$. This together with the edge effects due to the finite size of the system results in values for the correlation with $h_Z\neq0$ similar to values obtained with $h_Z=0$.
However, CDR and vnCDR do provide an advantageous mitigation with impressive visual results in some cases as shown in Fig.~\ref{fig:correlation_0} where we exhibit the correlation at $h_X=0.9$ and $h_Z=0$ with a Trotter step of $0.25$ which gives a final depth of $220$ at $tJ=5$, involving $320$ CNOT gates. The correlation with $h_Z=0.2$ is shown in Appendix~\ref{sec:data}. The dynamics which are almost entirely lost at late times are recovered to qualitative accuracy by vnCDR. In this case vnCDR mitigated results gave the lowest normalised absolute error. CDR also performed well giving very similar visual results. Showing that the dynamics of a complex observable can be qualitatively recovered for such deep circuits is a testament to the power of CDR and vnCDR.
\begin{figure}
\caption{Correlation of qubits at sites along the x axis with the central qubit for the TFIM with $h_Z=0$ and the system initialised in the $\ket{\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow}
\label{fig:correlation_0}
\end{figure}
Overall, we find that CDR and vnCDR lead to the best mitigated results for the observables explored in this work in almost all cases. The advantage is particularly clear for more complex observables. Interestingly, although vnCDR does generally have the smallest absolute error the advantage over CDR is slight. This could be attributed to how noise is being increased in the circuits of interest and when constructing the training set. More fine grained methods such as random identity insertions~\cite{He_2020} may be necessary to obtain a clear contrast between CDR and vnCDR. Alternatively, the lack of advantage in using multiple noise levels and near-Clifford training data could be due to the number of shots used to evaluate each circuit. More computational overhead might be necessary to obtain some improvement in the vnCDR results in comparison with CDR.
\section{Implementation details}
\subsection{Scaling the noise}
We perform the noise amplification in our quantum circuits using the so called fixed identity insertion method~\cite{He_2020}. We insert pairs of CNOT gates, which evaluate to the identity, after each CNOT implementation in the original circuit. Assuming the vast majority of error is introduced by these entangling gates, this method amplifies the noise by the factor of CNOT gates introduced. In our experiments we found it optimal to use noise levels $\mathcal{C} = \{1,3\}$ when using ZNE and vnCDR. Furthermore, a linear fit was used to extrapolate to the zero noise limit. We note that it would be interesting to implement a more fine grained noise amplification technique to explore if the results obtained by ZNE and vnCDR could improve. Additionally, more complex functions could be used to execute the extrapolation.
\begin{figure}
\caption{Distribution of exact and noisy magnetisation produced by the near-Clifford training circuits constructed using two different methods for time $t = 4$ ($8$ Trotter steps) with $h_X=0.5$, $h_Z=0.9$ and the initial state $\ket{\uparrow\uparrow\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\uparrow}
\label{fig:training_set_comparison1}
\end{figure}
\begin{figure*}
\caption{Evolution of the $\ket{\uparrow\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\uparrow\uparrow}
\label{fig:training_set_comparison2}
\end{figure*}
\subsection{Near-Clifford circuit training set}
Constructing the set of circuits that make up the training set is a key feature of both CDR and vnCDR. Intuitively, one desires a set of circuits close, in some sense, to the circuit of interest while also being diverse enough to accurately train the ansatz. In order to construct such a set of circuits we follow the protocol presented in Refs.~\cite{czarnik2020error, lowe2020unified}. In this work we restrict substitutions to a portion of the circuit beyond some depth.
First the circuit of interest is decomposed into the native gate set of the IBM quantum computers $\{R_{X}(\pi/2), R_{Z}(\theta), X, \text{CNOT}\}$. These gates are Clifford with the exception of $R_{Z}(\theta)$ which is only Clifford when $\theta = n \pi/2$, where $n = 0,1,2,3$ and correspond to the phase gate $S^{n}$. Therefore, we replace some of the $R_{Z}(\theta)$ gates by the phase gate to some power $n$.
Which gates (labelled $i$) to replace are chosen probabilistically according to distribution,
\begin{equation}
p(\theta_{i}) \propto \sum_{n=0}^3\exp(-||R_{Z}(\theta_{i})-S^{n}||^{2}/\sigma^{2}),
\end{equation}
where $||.||$ represents the Frobenius norm and sigma is a constant parameter taken here as $\sigma = 0.5$. Additionally, which Clifford gate to replace a chosen $R_{Z}(\theta)$ rotation with is also chosen probabilistically,
\begin{equation}
p'(n) \propto \exp(-||R_{Z}(\theta_{i})-S^{n}||^{2}/\sigma^{2}),
\end{equation}
also with $\sigma = 0.5$. We find this choice of $\sigma$ allows for construction of training sets which are diverse yet biased to the circuit of interest.
In both CDR and vnCDR implementations $50$ near Clifford circuits were constructed in this manner for each circuit of interest. Half the non-Clifford gates in each circuit were substituted, capped at $50$ non-Clifford gates.
Two approaches were compared: replacing gates throughout the entire circuit (method 1) and restricting the replacements to appear beyond a certain depth (method 2). We found that method 2 produces more similar observables to the circuit of interest, while still being sufficiently diverse. An example of a two CDR training sets constructed with both methods is shown in Fig.~\ref{fig:training_set_comparison1}. This example reflects the general trend observed, with training circuits being more similar to the circuit of interest when restricting Clifford substitutions to a fixed portion of the circuit. This kind of training set leads to a better mitigation for deeper circuits (see Fig.~\ref{fig:training_set_comparison2}).
This can be motivated by visualising a Clifford replacement as a unitary transformation on the original circuit. To minimise the action of this unitary one can imagine naively maximising the section of the circuit left unchanged, so forcing the Clifford substitution to appear as late as possible. We replace all non-Clifford gates in the second half of the circuit up to $50$ non-Clifford gates. Beyond $50$ non-Clifford gates we restrict all Clifford substitutions to appear at the greatest possible circuit depth. Fig.~\ref{fig:training_set_comparison2}$(c)$ shows the dispersion of each training set constructed by both methods at various circuit depths. We use a measure of dispersion to indicate the closeness of the training circuits to the circuit of interest, defined as:
\begin{equation}
\frac{1}{m} \sum_{i}^{m} \sqrt{(x_{i}-\langle X \rangle_{noisy})^{2} + (y_{i} - \langle X\rangle_{exact})^{2}},
\end{equation}
where $m$ is the number of training circuits and $x_{i}$, $y_{i}$ are the noisy, exact expectation values for the observable of interest for each of the training circuits and $\langle X \rangle_{noisy}$, $\langle X \rangle_{exact}$ are the noisy, exact expectation values for the circuit of interest.
In Fig.~\ref{fig:training_set_comparison2}$(b)$ an example of CDR is shown successfully mitigating noise in deep circuits, with this figure showing the dynamics of the magnetisation being recovered up to the final circuit depth of $176$. Oscillations in the magnetisation are recovered after they all but vanish from the raw data. The dispersion increases less quickly with circuit depth for method 2 than for method 1, shown in Fig.~\ref{fig:training_set_comparison2}$(c)$, suggesting method 2 makes for more reliable training sets. This is reflected in the more accurate mitigation results obtained.
In the case of pmCDR the training sets for the first two Trotter steps were used to train the model as outlined in Section \ref{sec:pmcdr}.
Once the circuits in the training set are executed (at two noise levels for each circuit of interest) this data is used to train the CDR and vnCDR ansatzes. We found for the majority of the observables investigated here the mitigation improved by repeatedly training the given anstaz on a randomly selected subset of the total training data. We used $200$ subsets with data from $5$ circuits each, taking the final mitigation as the median mitigated observables produced from each subset. We leave systematic investigation of this bootstrapped training method for a later work. All the observables of interest here can be calculated from the counts measured in the $Z$ basis. Therefore, data from the same training set from each circuit could be used to mitigate the noise on all the observables of interest.
\section{Conclusion}
In this work we have simulated the dynamics of a quantum quench on the TFIM using a trotterised evolution on a quantum computer. We applied several data-driven error mitigation techniques, as well as presenting a simplified implementation of CDR, so-called pmCDR inspired by Ref.~\cite{vovrosh2021efficient}. Using these techniques we have shown it is possible to calculate the first meson masses with quantitative accuracy for systems of $9$ qubits, the largest system explored in a study of this type. Clifford based mitigation methods show the best performance overall. We have demonstrated quantitative accuracy can be obtained using CDR and vnCDR from observables produced by circuits with depths of up to $176$ involving hundreds of CNOT gates. Furthermore, we have shown CDR and vnCDR enable the recovery of dynamics which appear completely washed out due to noise, highlighted in our measurements of the observable $\Delta_{i}^{ZZ}$ and the two-site correlation. pmCDR does work well consistently for shorter depth circuits, but begins to struggle as depth increases. A similar trend is observed for ZNE. Combining pmCDR with other mitigation strategies such as measurement error mitigation~\cite{funcke2020measurement}, random compilation and ZNE for the estimation of the noise parameters could improve its performance~\cite{urbanek2021}. In general CDR and vnCDR are advantageous due to the more general ansatzes fitted with training data which reflect the noise acting on the circuit of interest more accurately.
We have shown that making Clifford substitutions in a fixed region of the circuit of interest, beyond some depth, makes for a more accurate mitigation. The best training set construction method to use in general is still an open question. Clifford circuits are clearly useful mitigation strategies, but their performance could be enhanced with the development of well studied methods to construct a faithful training set. Furthermore, the exploration of more complex ansatzes is sure to provide promise in mitigating noise, as well as using training data suited to specific problems~\cite{montanaro2021error}. Finally, the combination of these methods with more recent error mitigation advances such as virtual distillation appears to be a promising research direction~\cite{czarnik2021qubitefficient}. It would also be interesting to explore recent variational algorithms~\cite{gibbs2021longtime} in conjunction with Clifford circuit based error mitigation to obtain some computationally non-trivial results.
Overall, improvement in quality of available mitigation techniques and quantum hardware becoming more widely accessible opens the possibility of near term useful quantum advantage. Near-Clifford circuit based mitigation methods are demonstrating their potential to become the staple error mitigation technique.
\section*{Acknowledgments}
We thank Piotr Czarnik for useful discussions. We also thank the IBM Quantum team for making devices available via the IBM Quantum Experience. The access to the IBM Quantum Experience has been provided by the CSIC IBM Q Hub. A.S.G is supported by the Spanish Ministry of Science and Innovation under grant number SEV-2016-0597-19-4. M.H.G is supported by “la Caixa” Foundation (ID 100010434),
Grant No. LCF/BQ/DI19/11730056. This work has also been financed by the Spanish grants PGC2018-095862-B-C21, QUITEMAD+ S2013/ICE-2801, SEV-2016-0597 of the ”Centro de Excelencia Severo Ochoa” Programme and the CSIC Research Platform on Quantum Technologies PTI-001.
\appendix
\section{CDR training set example\label{sec:CDR_cnst}}
We show an example of a Clifford training set which provides a much more faithful mitigation with the ansatz including a constant term (see Fig.~\ref{fig:training_set_CDR}). Using an ansatz which contains the constant clearly allows for more flexible fitting of the training data.
\begin{figure}
\caption{Distribution of exact and noisy magnetisation produced by the near-Clifford training circuits constructed for time $t = 3.75$ ($15$ Trotter steps) with $h_x=0.5$, $h_z=0.5$ and the initial state $\ket{\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow}
\label{fig:training_set_CDR}
\end{figure}
\section{Meson masses\label{sec:masses}}
In order to understand the phenomenon of confinement it is useful to project the Hamiltonian Eq.~\eqref{IisingHamiltonian} into the two kink subspace with basis $\{\ket{i,n}\}$:
\begin{align}
\mathcal{H}=&\sum_{i,n}\left[V(n)\ket{i,n}-Jh_X\left(\ket{i-1,n+1}\right.\right.\nonumber\\
&\phantom{\sum_{i,n}[}\left.\left.+\ket{i+1,n-1}+\ket{i,n-1}+\ket{i,n+1}\right)\right]\bra{i,n}
\label{eq:2-kink_H}
\end{align}
where $V(n)=2Jh_Zn$. The first term of this Hamiltonian represents a confining potential proportional to the separation between the domain walls and the second allows nearest neighbour interactions due to hopping. Therefore, a pair of kinks will experience an oscillatory motion due to the confining potential resulting in a meson.
In the case with $h_Z=0$, $\sigma^Z(t)$ decays to zero exponentially for any quench with $h_X<1$~\cite{calabrese_quantum_2011}. However, when a longitudinal field is introduced, the dynamics changes and an oscillatory behaviour is observed with various frequencies from which the masses of the two kinks bound states can be extracted. For this purpose, we consider the states $\ket{i,n}$ indicated in Eq.~\eqref{eq:meson} which are eigenstates of the Hamiltonian with $h_X=0$ and we perform a quench up to a certain value $h_X<1$. To obtain the eigenstates of the system after the quench we use the 2-kink model introduced before as it is a good approximation of the low energy behaviour of the system even for large values of $h_X$~\cite{tan2019observation}. To diagonalise the Hamiltonian Eq.~\eqref{eq:2-kink_H} we start by changing to the momentum space,
\begin{equation}
\ket{k,n}=\frac{1}{\sqrt{L-(n+1)}}\sum_j^{L-(n+1)}e^{-ikj-ik\frac{n}{2}}\ket{j,n},
\end{equation}
so that the Hamiltonian becomes
\begin{align}
\mathcal{H}=\sum_{k,n}&\left[V(n)\ket{k,n}\bra{k,n}+2h_X\cos\qty(\frac{k}{2})\right.\nonumber\\
&\left.\left(\ket{k,n}\bra{k,n+1}+\ket{k,n}\bra{k,n-1}\right)\right].
\end{align}
This Hamiltonian is diagonal in the basis of states
\begin{equation}
\ket{k,\alpha}=\sum_n\mathcal{C}_\alpha\mathcal{J}_{n-\nu_{k,\alpha}}\qty(x_k)\ket{k,n}
\end{equation}
where $\mathcal{J}$ is the Bessel function of the first kind, $\nu_{k,\alpha}=E_{k,\alpha}/2h_X$, $x_k=2h_Z\cos\qty(\frac{k}{2})/h_x$ and $\mathcal{C}_\alpha$ is a coefficient to normalise the state~\cite{vovrosh_confinement_2020}. Therefore, $\ket{k,\alpha}$ are the eigenstates of the Hamiltonian with $h_X\neq0$ and $h_Z\neq0$ and we can write the state of the system at time $t$ as
\begin{equation}
\ket{\Psi(t)}=\sum_{k,\alpha}\bra{k,\alpha}\ket{\Psi(0)}e^{-iE_{k,\alpha}t}\ket{k,\alpha}
\end{equation}
where $\ket{\Psi(0)}$ is the initial state. Using this expression, the expected value of a certain observable $\mathcal{O}$ is
\begin{align}
\bra{\Psi(t)}\mathcal{O}\ket{\Psi(t)}=\sum_{\substack{k,\alpha\\q,\beta}}&\bra{\Psi(0)}\ket{k,\alpha}\bra{q,\beta}\ket{\Psi(0)}\cdot\nonumber\\
&\cdot\bra{k,\alpha}\mathcal{O}\ket{q,\beta}e^{-i(E_{q,\beta}-E_{k,\alpha})t}
\label{eq:exp_val}
\end{align}
where we see an oscillatory behaviour with frequencies equal to energy differences between eigenstates. A method of obtaining the masses corresponding to different excited states, $m_\alpha=E_{0,\alpha}-E_{0,0}$, is to use initial states whose dominant oscillation frequency is $\omega_\alpha=E_{0,\alpha+1}-E_{0,\alpha}$ because then, the masses $m_\alpha$ are given by $m_\alpha=\sum_{\beta=0}^\alpha\omega_\beta$.
With the parameters $h_X$ and $h_Z$ that we are using, if we consider the initial states $\ket{i=4,n=1}=\ket{\uparrow\uparrow\uparrow\uparrow\downarrow\uparrow\uparrow\uparrow\uparrow}$ and $\ket{i=4,n=2}=\ket{\uparrow\uparrow\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\uparrow}$, the dominant frequencies are $\omega_1$ and $\omega_2$, respectively. The highest coefficients,
\begin{equation}
c_{k,\alpha,n}=\frac{\mathcal{C}_\alpha\mathcal{J}_{n-\nu_{k,\alpha}}}{\sqrt{L-(n+1)}},
\end{equation}
in the expansion Eq.~\eqref{eq:exp_val} written in the basis $\ket{j,n}$ are those corresponding to the states $\ket{k=0,\alpha=2}$ and $\ket{k=0,\alpha=1}$ for the initial state $\ket{i=4,n=1}$ and $\ket{k=0,\alpha=3}$ and $\ket{k=0,\alpha=2}$ for the initial state $\ket{i=4,n=2}$.
It should be noted that we cannot use the two-kink approximation to obtain the energy of states with $n=0$, such as the state $\ket{\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow}$. However, a dominant oscillation frequency is also observed in the temporal evolution of the magnetisation using this initial state. This oscillation frequency is to be understood as $\omega_0=E_{0,1}-E_{0,0}$ since it corresponds to the energy required to create a particle with momentum zero~\cite{PhysRevA.95.023621}.
Following this prescription, we show in Fig.~\ref{fig:mass_0.5} the masses corresponding to the frequencies of Fig.~\ref{fig:freq_0.5}.
\section{Additional data \label{sec:data}}
Here we present the results for the time evolution of the local magnetisation with different values of $h_Z$ used to obtain the frequencies shown in Fig.~\ref{fig:freq_0.5}. (see Fig.~\ref{fig:magnetisation_0.5-0.65-0.9}). We also show a colour plot of the magnetisation for each qubit in the system for $h_{X}=0.5, h_{Z} = 0.9$ (see Fig.~\ref{fig:magnetisation_0.9_Q9}). Furthermore, we show the additional results for the correlation when $h_{Z} = 0.2$ (see Fig.~\ref{fig:correlation_0.2}). The mitigation of the correlation here is a little worse than that presented in the main text. This could be attributed to the number of non-Clifford gates per Trotter step being greater when $h_{Z} \neq 0$. This means for the same circuit depth more gates need to be substituted when forming the training circuits, making for a less reliable dataset in general.
\begin{figure}
\caption{Correlation of qubits at sites along the x axis with the central qubit for the TFIM with $h_Z=0.2$ and the system initialised in the $\ket{\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow}
\label{fig:correlation_0.2}
\end{figure}
\begin{figure}
\caption{Masses obtained at $h_X=0.5$ and different longitudinal fields from the exact diagonalised (dashed lines), simulated (a), raw (b) and mitigated data (c)-(f). Masses obtained for initial states $\ket{\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow}
\label{fig:mass_0.5}
\end{figure}
\section{Quantum circuit for trotterised evolution\label{sec:circuit}}
The quantum circuit to evolve the system by one Trotter step for a $5$ spin system is shown in Fig.~\ref{fig:QCircuit}. In general for a system with an odd number of qubits $Q$ each Trotter step has a depth of $11$ with $2(Q-1)$ entangling gates and $3Q-1$ non-Clifford gates.
\pagebreak
\section{Data collection}
We used the Paris quantum computers available through the IBM quantum cloud access. The data in this work was collected during the days between 22nd February 2021 and 5th March 2021 the errors in the Paris machine remained consistent over the times of collection and are shown in Fig.~\ref{fig:computer_errors}. We randomly selected the qubits to form a $9$ qubit chain. The results obtained here could be improved by choosing the qubits to use in a systematic manner, minimising the single qubit and CNOT error.
\begin{figure*}
\caption{Temporal evolution of $Z$-axis local magnetisation with $h_X=0.5$ and $h_Z=0.5,\text{ }
\label{fig:magnetisation_0.5-0.65-0.9}
\end{figure*}
\begin{figure*}
\caption{Temporal evolution of $Z$-axis local magnetisation with $h_X=0.5$ and $h_Z=0.9$ for the initial state $\ket{\uparrow\uparrow\uparrow\uparrow\downarrow\uparrow\uparrow\uparrow\uparrow}
\label{fig:magnetisation_0.9_Q9}
\end{figure*}
\begin{figure*}
\caption{ Quantum circuit representation showing one step of a first order Trotter expansion of a $5$-qubit encoded spin system.}
\label{fig:QCircuit}
\end{figure*}
\begin{figure*}
\caption{Errors on the IBMQ Paris quantum computer at the time the circuits in this work were run. The qubits we used to form a spin chain are encircled in red. For the qubits used in this work $T_{1}
\label{fig:computer_errors}
\end{figure*}
\end{document} | math |
تٔمۍ کٔر پننہِ کیریئرٕچ شروعات أکِس مرکزی اداکار سنٛدۍ پٲٹھۍ حالانکہ سُہ روُد معاون کردارن تہٕ مُخٲلفن پٲٹھۍ زیادٕ کامیاب | kashmiri |
ಹೊಸಪೇಟೆ: ಸ್ವತಃ ಕಟೌಟ್ ತೆರವುಗೊಳಿಸಿದ ಸಚಿವ ಆನಂದ್ ಸಿಂಗ್ ಹೊಸಪೇಟೆಡಿ.05: ನಗರದಲ್ಲಿನ ಅನಧಿಕೃತ ಬ್ಯಾನರ್ ಹಾಗೂ ಜಿಲ್ಲಾ ಉಸ್ತುವಾರಿ ಸಚಿವರ ಕಟೌಟ್ ಅನ್ನು ತೆರವುಗೊಳಿಸುವಂತೆ ಒತ್ತಾಯಿಸಿ ನಗರದಲ್ಲಿ ಶುಕ್ರವಾರ ರಾತ್ರಿ ಕರ್ನಾಟಕ ರಾಷ್ಟ್ರ ಸಮಿತಿ ಪಕ್ಷದ ರಾಜ್ಯಾಧ್ಯಕ್ಷ ರವಿ ಕೃಷ್ಣಾರೆಡ್ಡಿ ನೇತೃತ್ವದಲ್ಲಿ ಧ್ವಜಸ್ತಂಭ ವೃತ್ತದ ಬಳಿ ಧರಣಿ ನಡೆಸಿದರು. ಈ ವೇಳೆ ಸ್ಥಳಕ್ಕೆ ಆಗಮಿಸಿದ ಸಚಿವ ಆನಂದ್ ಸಿಂಗ್ ಸ್ವತಃ ಕಟೌಟ್ ತೆರವುಗೊಳಿಸಿದರು. ಕಿತ್ತೂರಿನಿಂದ ಬಳ್ಳಾರಿ ವರೆಗೆ ಚಲಿಸು ಕರ್ನಾಟಕ ಎಂಬ 300 ಕಿಮೀ ಸೈಕಲ್ ಯಾತ್ರೆ ಹಮ್ಮಿಕೊಂಡಿದ್ದ ಪಕ್ಷವು ಶುಕ್ರವಾರ ನಗರಕ್ಕೆ ಆಗಮಿಸಿತ್ತು. ಇದೇ ವೇಳೆ ನಗರದಲ್ಲಿ ಅನಧಿಕೃತವಾಗಿ ಸಂಚಾರಿ ಸಿಗ್ನಲ್ ಕಾಣದಂತೆ ಹಾಕಿದ್ದ ಹುಟ್ಟುಹಬ್ಬ ಶುಭಾಶಯದ ಬ್ಯಾನರ್ ಅನ್ನು ಯಶಸ್ವಿಯಾಗಿ ತೆರವುಗೊಳಿಸಿದ ಆನಂತರ ನಗರದ ಶಾನ್ಭಾಗ್ ವೃತ್ತದಲ್ಲಿ ಹಾಕಿರುವ ಸಚಿವ ಆನಂದ್ ಸಿಂಗ್ ಅವರ ಕಟೌಟ್ ಅನ್ನು ಸಹ ನಗರಸಭೆ ತೆರವುಗೊಳಿಸುವಂತೆ ಒತ್ತಾಯಿಸಿ ಕಟೌಟ್ ಎದುರು ಧರಣಿ ನಡೆಸಿದರು. ಬಳ್ಳಾರಿ ವಿಭಜನೆ : ಕನ್ನಡದ ಮೇಲೇನು ಪರಿಣಾಮ? ಸ್ಥಳಕ್ಕೆ ಆಗಮಿಸಿದ ಸಚಿವ ಆನಂದ್ ಸಿಂಗ್ ಅವರು ಕಟೌಟ್ ತೆರವುಗೊಳಿಸಲಾಗುವುದು. ಕಾನೂನಿಗೆ ನಾವು ಗೌರವ ಕೊಡುತ್ತೇವೆ. ಮುಂದಿನ ಪೀಳಿಗೆಗೆ ಗೌರವ ಕೊಡುವ ಮಾದರಿಯಲ್ಲಿ ನಾವು ನಡೆದುಕೊಳ್ಳಬೇಕು. ಏಕವಚನದಲ್ಲಿ ಯಾರಿಗೂ ಮಾತನಾಡಬಾರದು. ಯಾರ ಮನಸ್ಸು ನೋವಿಸಬಾರದು ಎಂದು ರವಿಕೃಷ್ಣ ರೆಡ್ಡಿ ಅವರ ಬಳಿ ಹೇಳಿದರು. ಆಗ ರವಿ ಕೃಷ್ಣರೆಡ್ಡಿ ಅವರು ಕೂಡ ನಾವು ಹಮ್ಮಿಕೊಂಡ ಜಾಥಾ ಉದ್ದೇಶವೇ ಬೇರೆ ಇದೆ. ಆದರೆ, ಈ ರೀತಿ ಪ್ರಸಂಗ ನಡೆಯಿತು. ನಡೆಯಬಾರದಿತ್ತು ಎಂದು ಸಚಿವರ ಬಳಿ ಹೇಳಿದರು. ಕರ್ನಾಟಕ ರಾಷ್ಟ್ರಸಮಿತಿ ಪಕ್ಷದ ಕಾರ್ಯಕ್ರಮ ಮುಗಿಯುತ್ತಿದ್ದಂತೆಯೇ ಸ್ವತಃ ಸಚಿವ ಆನಂದ್ ಸಿಂಗ್ ಕಟೌಟ್ ತೆರವುಗೊಳಿಸಿದರು. | kannad |
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تینوں کمپنیوں کوچارارب 89 کروڑ 80 لاکھ روپے فی لائسنس ادا کرنا ہوں گے فوٹو ایکسپریس اسلام بادسپریم کورٹ کی مشروط اجازت کے بعد پاکستان کے پہلے ڈی ٹی ایچ ڈائریکٹ ٹوہوم لائسنس نیلام کیے گئے ایکسپریس نیوزکے مطابق اسلام اباد میں سپریم کورٹ کی مشروط اجازت کےبعد پاکستان کے پہلے ڈی ٹی ایچ ڈائریکٹ ٹوہوم لائسنس 14 ارب 69 کروڑ 40 لاکھ میں نیلام کیےگئے نیلامی کا غاز گزشتہ روز دوپہر 12 بجے 20 کروڑ کی ابتدائی قیمت سے ہوا اور بولی کاعمل 15 سے زائد گھنٹےتک جاری رہنے کے بعد اج صبح چار بجے سے پہلے ختم ہوا میگ انٹرٹینمنٹ لاہورنے ارب 91 کروڑروپے شہزاد اسکائے نے ارب 90 کروڑروپے جب کہ اسٹار ٹائمز کمیونیکیشن نے4 ارب 89 کروڑ 80 لاکھ روپے کی بولی لگا کر لائسنسن اپنے نام کیےقوائد کے مطابق سب سے زیادہ بولی دینے والی دو کمپنیاں بھی تیسرے نمبر پر بولی دینے والی کمپنی کے مساوی رقم ادا کریں گی اس طرح تینوں کمپنیوں کو چار ارب 89 کروڑ 80 لاکھ روپے فی لائسنس ادا کرنا ہوں گے جب کہ سیکیورٹی کلیئرنس کےبعد کمپنیوں کوڈی ٹی ایچ لائسنس دے دیے جائیں گےلائسنس کی فیس پہلے 15 فیصد پھر دوسری قسط 50 اور تیسری 35 فیصد جمع کرانا ہوگی واضح رہے کہ لاہورہائی کورٹ نے ڈی ٹی ایچ کی نیلامی کے خلاف حکم امتناعی جاری کیا تھا جسے گذشتہ روز پیمرا کی قانونی ٹیم نے سپریم کورٹ کے سامنے پیش کیا جس پرسپریم کورٹ نے حکم امتناعی کو خارج کرتے ہوئے نیلامی جاری رکھنے کا حکم دیا تھا | urdu |
\begin{document}
\begin{abstract}
We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis.
We also observe that if $V$ is a model of $\mathsf{ZFC}$, then $\mathsf{DC_{<\kappa}}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $\delta<\kappa$ and $V$ is a model of $\mathsf{ZF+DC_{\delta}}$, then $\mathsf{DC_{\delta}}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is ($\delta+1$)-strategically closed and $\mathcal{F}$ is $\kappa$-complete.
\end{abstract}
\maketitle
\section{introduction}
Grigorieff \cite{Gri1975} proved that symmetric extensions in terms of {\em symmetric system} $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$\footnote{ $\langle\mathbb{P},\mathcal{G},\mathcal{F}\rangle$ is a {\em symmetric system} if $\mathbb{P}$ is a forcing notion, $\mathcal{G}$ is a group of automorphisms of $\mathbb{P}$, and $\mathcal{F}$ is a normal filter of subgroups over $\mathcal{G}$.} are intermediate models of the form HOD$(V[a])^{V[G]}$ as $a$ varies over $V[G]$. Apter, Henle, Cody, and Koepke constructed several models of $\mathsf{ZF}$ in terms of hereditarily definable sets based on L\'{e}vy Collapse (cf. \cite{AC2013, Apt1983a, AK2006, Apt2005, AH1991}). The purpose of this note is to translate the arguments of a few of those choiceless model constructions to symmetric extensions in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$ and extend a few published results. In particular, we prove the following.
\begin{enumerate}
\item We prove the failure of $\mathsf{AC_{\kappa}}$ (Every family of $\kappa$ non-empty sets admits a choice function) in the symmetric extension of \cite[\textbf{Theorem 4.1}]{Kar2019}. Moreover, we study an argument to preserve the supercompactness of $\kappa$ in the symmetric model following the methods of \cite{Ina2013}.
\item We reduce the large cardinal assumption of \cite[\textbf{Theorem 2}]{AC2013} and \cite[\textbf{Theorem 3}]{AC2013}.
\item We observe an {\em infinitary Chang conjecture} in the choiceless model constructed in \cite[\textbf{Theorem 11}]{AK2006}. Moreover, we prove that $\aleph_{\omega_{1}}$ is an almost Ramsey cardinal in the model.
\item Fix an arbitrary $n_{0}\in \omega$. We observe that if $\langle S_{k}: 1\leq k <\omega\rangle$ is a sequence of stationary sets such that $S_{k}\subseteq\aleph_{n_{0}+2(k+1)}$ for every $1\leq k <\omega$, then $\langle S_{k}: 1\leq k <\omega\rangle$ is {\em mutually stationary} in the choiceless model constructed in \cite[\textbf{Theorem 1}]{Apt1983a}. We also observe an alternating sequence of measurable and non-measurable cardinals in the model. Moreover, we observe that $\aleph_{\omega}$ is an almost Ramsey cardinal in the model.
\end{enumerate}
Secondly, we prove a conjecture of Dimitriou related to the failure of Dependent Choice--or $\mathsf{DC}$--in a symmetric extension based on finite support products of collapsing functions, from \cite{Dim2011}. We also study new lemmas related to preserving $\mathsf{DC}$ in symmetric extensions inspired by \cite[\textbf{Lemma 1}]{Kar2014}. In particular, we observe the following.
\begin{enumerate}
\item Let $V$ be a model of $\mathsf{ZFC}$. If $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete, then $\mathsf{DC_{<\kappa}}$ is preserved in the symmetric extension of $V$ with respect to the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$.
\item Let $\delta<\kappa$ and $V$ be a model of $\mathsf{ZF+DC_{\delta}}$ where the Axiom of Choice ($\mathsf{AC}$) might fail. If $\mathbb{P}$ is ($\delta+1$)-strategically closed and $\mathcal{F}$ is $\kappa$-complete, then $\mathsf{DC_{\delta}}$ is preserved in the symmetric extension of $V$ with respect to the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$.
\end{enumerate}
\subsection{Preserving Dependent Choice} Karagila \cite[\textbf{Lemma 1}]{Kar2014} proved that if $\mathbb{P}$ is $\kappa$-closed and $\mathcal{F}$ is $\kappa$-complete then $\mathsf{DC_{<\kappa}}$ is preserved in the symmetric extension in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$.
We observe that `$\mathbb{P}$ is $\kappa$-closed' can be replaced by `$\mathbb{P}$ is $\kappa$-distributive' in \cite[\textbf{Lemma 1}]{Kar2014}. This slightly generalize \cite[\textbf{Lemma 1}]{Kar2014}, since there are $\kappa$-strategically closed forcing notions which are not $\kappa$-closed\footnote{As for an example, the forcing notion $\mathbb{P}(\kappa)$ which adds a {\em non-reflecting stationary set of cofinality $\omega$ ordinals in $\kappa$}, is $\kappa$-strategically closed but not even $\omega_{2}$-closed. (cf. \cite[\textbf{section 6}]{Cum2010}).} and $\kappa$-distributivity is weaker than $<\kappa$-strategic closure.\footnote{As for an example, the forcing notion for {\em killing a stationary subset of $\omega_{1}$}, is $\omega_{1}$-distributive but not even $<\omega_{1}$-strategically closed (cf. \cite[\textbf{section 6}]{Cum2010}).}
\begin{observation}{(\textbf{Lemma 3.2})}
{\em Let $V$ be a model of $\mathsf{ZFC}$. If $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete, then $\mathsf{DC_{<\kappa}}$ is preserved in the symmetric extension of $V$ with respect to the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$.}
\end{observation}
We also observe that even if we start with a model $V$, which is a model of $\mathsf{ZF +DC_{\delta}}$ where $\mathsf{AC}$ might fail, we can still preserve $\mathsf{DC_{\delta}}$ in a symmetric extension of $V$ in certain cases. In particular, we observe the following.
\begin{observation}{(\textbf{Lemma 3.4})}
{\em Let $\delta<\kappa$ and $V$ be a model of $\mathsf{ZF + DC_{\delta}}$. If $\mathbb{P}$ is ($\delta+1$)-strategically closed and $\mathcal{F}$ is $\kappa$-complete, then $\mathsf{DC_{\delta}}$ is preserved in the symmetric extension of $V$ with respect to the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$. }
\end{observation}
\subsection{On a question of Apter} Woodin asked in the context of $\mathsf{ZFC}$, that if $\kappa$ is strongly compact and $\mathsf{GCH}$ holds below $\kappa$, then must $\mathsf{GCH}$ hold everywhere? The problem is still open in the context of $\mathsf{ZFC}$. One variant of this question is if $\mathsf{GCH}$ can fail at every limit cardinal less than or equal to a strongly compact cardinal $\kappa$ where as $\mathsf{GCH}$ holds above $\kappa^{+}$. Apter answered this in the context of $\mathsf{ZF}$. Apter \cite[\textbf{Theorem 3}]{Apt2012} constructed a model where $\kappa$ is a regular limit cardinal and a supercompact cardinal, and $\mathsf{GCH}$ holds for a limit $\delta$ if and only if $\delta>\kappa$. In that model the Countable Choice--or $\mathsf{AC_{\omega}}$-- fails. At the end of \cite{Apt2012}, Apter asked the following question.
\begin{question}
{\em Is it possible to construct analogs of Theorem 3 in which some weak version of $\mathsf{AC}$ holds ?}
\end{question}
The author and Karagila constructed a symmetric extension to answer \textbf{Question 1.3} in \cite[\textbf{Theorem 4.1}]{Kar2019}.
\begin{thm}{(cf. \cite[\textbf{Theorem 4.1}]{Kar2019})}
{\em Let V be a model of $\mathsf{ZFC + GCH}$ with a supercompact cardinal $\kappa$. Then there is a
symmetric extension in which $\mathsf{DC_{<\kappa}}$ holds, $\kappa$ is a regular limit cardinal and supercompact, and $\mathsf{GCH}$ holds for a limit cardinal $\delta$ if and only if $\delta >\kappa$.}
\end{thm}
For the sake of convenience, we call the symmetric extension constructed
in \cite[\textbf{Theorem 4.1}]{Kar2019}
as $\mathcal{N}_{1}$. We study an argument to preserve the supercompactness of $\kappa$ in $\mathcal{N}_{1}$ applying the methods of \cite{Ina2013} and prove the following in \textbf{section 4}.
\begin{thm}
{\em In $\mathcal{N}_{1}$, $\mathsf{AC_{\kappa}}$ fails.}
\end{thm}
\subsection{Proving Dimitriou's conjecture} Dimitriou constructed a symmetric extension based on finite support products of collapsing functions. At
the end of \cite[\textbf{section 1.4}]{Dim2011}, Dimitriou conjectured that $\mathsf{DC}$ would fail in the symmetric extension (cf. \cite[\textbf{Question 1}, \textbf{Chapter 4}]{Dim2011}).
We prove the conjecture. For the sake of convenience, we call this model as Dimitriou's model and prove the following in \textbf{section 5}.
\begin{thm}
{\em In Dimitriou's model, $\mathsf{AC_{\omega}}$ fails.}
\end{thm}
\subsection{Reducing the assumption of supercompactness to strong compactness} Apter and Cody \cite[\textbf{Theorem 2}]{AC2013} obtained a model of $\mathsf{ZF +\neg AC_{\omega}}$ where $\aleph_{1}$ and $\aleph_{2}$ are both singular of cofinality $\omega$, and there is a sequence of distinct subsets of $\aleph_{1}$ of length equal to any predefined ordinal, assuming a supercompact cardinal $\kappa$. In \textbf{section 6}, we observe that applying a recent result of Usuba (cf. \cite[\textbf{Theorem 3.1}]{ADU2021}) followed by working with a model of $\mathsf{ZF +\neg AC_{\omega}}$ constructed using {\em strongly compact Prikry forcing}
, it is possible to reduce the assumption of a supercompact cardinal $\kappa$ to a strongly compact cardinal $\kappa$.
\begin{observation}
{\em Suppose that $\kappa$ is a strongly compact cardinal, $\mathsf{GCH}$ holds, and $\theta$ is an ordinal. Then there is a model of $\mathsf{ZF +\neg AC_{\omega}}$ in which $cf(\aleph_{1})=cf(\aleph_{2})=\omega$,
and there is a sequence of distinct subsets of $\aleph_{1}$ of length $\theta$.}
\end{observation}
Similarly, we reduce the large cardinal assumption of \cite[\textbf{Theorem 3}]{AC2013} from a supercompact cardinal to a strongly compact cardinal. Apter and Cody \cite[\textbf{Theorem 3}]{AC2013} obtained a model of $\mathsf{ZF +\neg AC_{\omega}}$ where $\aleph_{\omega}$ and $\aleph_{\omega+1}$ are both singular with $\omega\leq cf(\aleph_{\omega+1})<\aleph_{\omega}$, and there is a sequence of distinct subsets of $\aleph_{\omega}$ of length equal to any predefined ordinal, assuming a supercompact cardinal $\kappa$.
We prove the following in \textbf{section 6}.
\begin{observation}
{\em Suppose that $\kappa$ is a strongly compact cardinal, $\mathsf{GCH}$ holds, and $\theta$ is an ordinal. Then there is a model of $\mathsf{ZF +\neg AC_{\omega}}$ in which $\aleph_{\omega}$ and $\aleph_{\omega+1}$ are both singular with $\omega\leq cf(\aleph_{\omega+1})<\aleph_{\omega}$, and there is a sequence of distinct subsets of $\aleph_{\omega}$ of length $\theta$.}
\end{observation}
\subsection{Infinitary Chang conjecture from a measurable cardinal} Assuming a measurable cardinal, Apter and Koepke constructed a model $\mathcal{N}$ of $\mathsf{ZF}$ based on L\'{e}vy collapse in \cite[\textbf{Theorem 11}]{AK2006}. In $\mathcal{N}$, $\omega_{1}$ is singular,
and $\aleph_{\omega_{1}}$ is a Rowbottom cardinal carrying a Rowbottom filter. They mentioned that in $\mathcal{N}$, $\mathsf{AC_{\omega}}$ fails because of the singularity of $\omega_{1}$. In \textbf{section 7}, we observe an {\em infinitary Chang conjecture} in a symmetric extension, which is very similar to $\mathcal{N}$, except we consider a finite support product construction. We use the observation that it is possible to force a {\em coherent} sequence of Ramsey cardinals after performing Prikry forcing on a normal measure over a measurable cardinal $\kappa$ (cf. \cite[\textbf{Theorem 3}]{AK2006}). We also use the observation that an infinitary Chang conjecture can be established in a symmetric model, assuming a coherent sequence of Ramsey cardinals. As in the model of \cite[\textbf{Theorem 11}]{AK2006}, $\omega_{1}$ is singular and therefore $\mathsf{AC_{\omega}}$ fails.
\begin{thm}
{\em Let $V'$ be a model of $\mathsf{ZFC}$ where there is a measurable cardinal. Then there is a generic extension $V$ of $V'$, and a symmetric extension $V(G)$ of $V$ such that $\omega_{1}$ is singular in $V(G)$.
Moreover, an infinitary Chang conjecture holds in $V(G)$.
}
\end{thm}
Similarly, we also observe an infinitary Chang conjecture in the model $\mathcal{N}$. For the sake of convenience, we call the model $\mathcal{N}$ as Apter and Koepke's model and prove the following in \textbf{section 7}.
\begin{thm}
{\em An infinitary Chang conjecture holds in Apter and Koepke's model. Moreover, $\aleph_{\omega_{1}}$ is an almost Ramsey cardinal in the model.}
\end{thm}
\subsection{Mutual stationarity property from a sequence of measurable cardinals}
Foreman and Magidor \cite{FM2001} introduced the idea of {\em mutual stationarity}. They asked if there is a model of set theory in which every sequence of stationary subsets of the $\aleph_{n}$'s of a fixed cofinality is mutually
stationary (cf. \cite[page 290]{FM2001}). Assuming an $\omega$-sequence of supercompact cardinals, Apter \cite[\textbf{Theorem 1}]{Apt2005} constructed a model of $\mathsf{ZF}+\mathsf{DC}$ in which if $\langle S_{n} : 1 \leq n<\omega\rangle$ is a sequence of stationary sets such that $S_{n} \subseteq \aleph_{n}$ for every $1 \leq n <\omega$, then $\langle S_{n} : 1 \leq n<\omega\rangle$ is mutually stationary. Apter \cite[\textbf{Theorem 1}]{Apt1983a} further obtained a similar model based on L\'{e}vy collapse as constructed in \textbf{\cite{Apt2005}}, where $\aleph_{\omega}$ carries a Rowbottom filter and $\mathsf{DC_{\aleph_{n_{0}}}}$ holds for any arbitrary $n_{0}\in \omega$, from an $\omega$-sequence of measurable cardinals. For the sake of convenience, we fix an arbitrary $n_{0}\in \omega$ in the ground model $V$, call the model from \cite[\textbf{Theorem 1}]{Apt1983a} as $\mathcal{N}_{n_{0}}$, and prove the following in \textbf{section 8}.
\begin{observation}
{\em The following hold in the model $\mathcal{N}_{n_{0}}$.
\begin{enumerate}
\item For each $1\leq k<\omega$, $\aleph_{n_{0}+2(k+1)}$ is a measurable cardinal and $\aleph_{n_{0}+2k+1}$ is not a measurable cardinal. In particular, for each $1\leq k<\omega$, there are no unifrom ultrafilters on $\aleph_{n_{0}+2k+1}$.
\item If $\langle S_{k}: 1\leq k <\omega\rangle$ is a sequence of stationary sets such that $S_{k}\subseteq\aleph_{n_{0}+2(k+1)}$ for every $1\leq k <\omega$, then $\langle S_{k}: 1\leq k <\omega\rangle$ is mutually stationary.
\item $\aleph_{\omega}$ is an almost Ramsey cardinal.
\end{enumerate}
}
\end{observation}
\textbf{Structure of the paper.}
\begin{itemize}
\item In \textbf{section 2}, we cover the basics.
\item In \textbf{section 3}, we prove \textbf{Observations 1.1 $\&$ 1.2}.
\item In \textbf{section 4}, we prove \textbf{Theorem 1.5}.
\item In \textbf{section 5}, we prove \textbf{Theorem 1.6}.
\item In \textbf{section 6}, we prove \textbf{Observations 1.7 $\&$ 1.8}.
\item In \textbf{section 7}, we prove \textbf{Theorems 1.9 $\&$ 1.10}.
\item In \textbf{section 8}, we prove \textbf{Observation 1.11}.
\end{itemize}
\section{Basics}
\subsection{Large Cardinals}
In this section, we recall the definition of inaccessible cardinals in the context of $\mathsf{ZFC}$ and other large cardinals in the context of $\mathsf{ZF}$. In $\mathsf{ZFC}$, we say $\kappa$ is a strongly inaccessible cardinal if it is a regular strong limit cardinal where the definition of ``strong limit" is that for all $\alpha<\kappa$, we have $2^{\alpha}<\kappa$. In the context of $\mathsf{ZF}$, the above definition doesn't make sense, as $2^{\alpha}$ may not be well-ordered. We refer the reader to \cite{BDL2007} for details concerning inaccessible cardinals in the context of $\mathsf{ZF}$.
We recall some large cardinal definitions in the context of $\mathsf{ZF}$ from \cite{Kan2003}.
\begin{defn} Let $\kappa$ be an uncountable cardinal.
\begin{enumerate}
\item The cardinal $\kappa$ is {\em weakly compact} if for all $f:[\kappa]^{2}\rightarrow 2$, there is a homogeneous set $X\subseteq \kappa$ for $f$ of order type $\kappa$.
\item The cardinal $\kappa$ is {\em Ramsey} if for all $f:[\kappa]^{<\omega}\rightarrow 2$, there is a homogeneous set $X\subseteq \kappa$ for $f$ of order type $\kappa$.
\item The cardinal $\kappa$ is {\em almost Ramsey} if for all $\alpha<\kappa$ and $f:[\kappa]^{<\omega}\rightarrow 2$, there is a homogeneous set $X\subseteq \kappa$ for $f$ having order type $\alpha$.
\item The cardinal $\kappa$ is {\em $\mu$-Rowbottom} if for all $\alpha<\kappa$ and $f:[\kappa]^{<\omega}\rightarrow \alpha$, there is a homogeneous set $X\subseteq\kappa$ for $f$ of order type $\kappa$ such that $\vert f^{''}[X]^{<\omega}\vert<\mu$. We say that $\kappa$ is Rowbottom if it is $\omega_{1}$-Rowbottom. A filter $\mathcal{F}$ on $\kappa$ is a {\em Rowbottom filter} on $\kappa$ if for any $f:[\kappa]^{<\omega}\rightarrow\lambda$, where $\lambda<\kappa$, there is a set $X\in\mathcal{F}$ such that $\vert f^{''}[X]^{<\omega}\vert\leq\omega$.
\item The cardinal $\kappa$ is {\em measurable} if there is a $\kappa$-complete free ultrafilter on $\kappa$.
A filter $\mathcal{F}$ on a cardinal $\kappa$ is {\em normal} if it is closed under diagonal intersections:
\begin{center}
If $X_{\alpha}\in \mathcal{F}$ for all $\alpha<\kappa$, then $\Delta_{\alpha<\kappa}X_{\alpha}\in \mathcal{F}$.
\end{center}
In $\mathsf{ZF}$ we have the following lemma.
\begin{lem}{(cf. \cite[\textbf{Lemma 0.8}]{Dim2011})}
{\em An ultrafilter $\mathcal{U}$ over $\kappa$ is normal if and only if for every regressive $f:\kappa\rightarrow\kappa$ there is an $X\in \mathcal{U}$ such that $f$ is constant on $X$.}
\end{lem}
Thus, we say an ultrafilter $\mathcal{U}$ over $\kappa$ is normal if for every regressive $f:\kappa\rightarrow\kappa$ there is an $X\in \mathcal{U}$ such that $f$ is constant on $X$.
\item For a set $A$, we say $\mathcal{U}$ is a {\em fine measure} on $\mathcal{P}_{\kappa}(A)$ if $\mathcal{U}$ is a $\kappa$-complete ultrafilter and for any $i\in A$, $\{x\in\mathcal{P}_{\kappa}(A): i\in x\}\in\mathcal{U}$. We say that $\mathcal{U}$ is a {\em normal measure} on $\mathcal{P}_{\kappa}(A)$ if $\mathcal{U}$ is a fine measure and if $f:\mathcal{P}_{\kappa}(A)\rightarrow A$ is such that $f(X)\in X$ for a set in $\mathcal{U}$, then $f$ is constant on a set in $\mathcal{U}$. The cardinal $\kappa$ is {\em $\lambda$-strongly compact} if there is a fine measure on $ \mathcal{P}_{\kappa}(\lambda)$; it is {\em strongly compact} if it is $\lambda$-strongly compact for all $\kappa\leq\lambda$.
\item The cardinal $\kappa$ is {\em $\lambda$-supercompact} if there is a normal measure on $ \mathcal{P}_{\kappa}(\lambda)$; it is {\em supercompact} if it is $\lambda$-supercompact for all $\kappa\leq\lambda$.
\end{enumerate}
\end{defn}
\begin{remark}
We note that the definition of supercompact (similarly strongly compact) is given in the terms of ultrafilters, which is weaker than the definition of supercompact in terms of elementary embedding due to Woodin \cite[\textbf{ Definition 220}]{Wood2010} (e.g. $\aleph_{1}$ can be supercompact or strongly compact if we consider the definition of supercompact or strongly compact in terms of ultrafilters, but $\aleph_{1}$ can not be the critical point of an elementary embedding (cf. \cite{Ina2013})).
\end{remark}
\begin{remark}
Ikegami and Trang \cite[\textbf{section 2}]{IT2019} defined that an ultrafilter $\mathcal{U}$ on $\mathcal{P}_{\kappa}X$ is normal if for any set $A \in \mathcal{U}$ and $f : A \rightarrow \mathcal{P}_{\kappa}X$
with $\emptyset \not= f(\sigma) \subseteq \sigma$ for all $\sigma\in A$, there is an $x_{0} \in X$ such that for $\mathcal{U}$-measure one many $\sigma$ in $A$, $x_{0}\in f(\sigma)$. They note that their definition of normality is equivalent to the closure under diagonal intersections in $\mathsf{ZF}$, while it may not be equivalent to the definition of normality in our sense without $\mathsf{AC}$.
\end{remark}
From now on, all our inaccessible cardinals are strongly inaccessible. We recall that a limit of Ramsey cardinals is an almost Ramsey cardinal in $\mathsf{ZF}$ (cf. \cite[\textbf{Proposition 1}]{AK2008}).
\subsection{Forcing extension and L\'{e}vy--Solovay Theorem}
Let $\mathbb{P}$ be a forcing notion, by which we mean a partially ordered set with a
maximum element 1. For $p, q \in \mathbb{P}$, we say that $p$ is stronger than $q$ or $p$ extends $q$ if $p \leq q$. Let $G$ be a $\mathbb{P}$-generic filter over $V$ and
$V^{\mathbb{P}}$ be a class of all $\mathbb{P}$-names defined recursively as follows: if $\tau$ is a set, $\tau \in V^{\mathbb{P}}$ if and only if $\tau \subseteq V^{\mathbb{P}} \times \mathbb{P}$. The interpretation of a $\mathbb{P}$-name $\tau$ by $G$ is defined recursively as $\tau^G = \{\sigma ^{G} : \exists p \in G((\sigma, p) \in \tau)\}$ and the generic
extension is defined as $V[G]=\{\tau^{G} : \tau \in V^{\mathbb{P}}\}$.
We recall that $V[G]$ is the smallest transitive
model of $\mathsf{ZFC}$ which has the same ordinals as $V$ and contains both $V$ and $G$. If $\mathbb{P}$ is our forcing notion and $G$ is a $\mathbb{P}$-generic filter over $V$, we will abuse notation somewhat and use both $V^{\mathbb{P}}$ and $V[G]$ to denote the generic extension of $V$.
We state a part of L\'{e}vy--Solovay Theorem (cf. \cite[\textbf{Theorem 21.2}]{Jec2003}) in $\mathsf{ZFC}$.
\begin{thm}
{\em
Let $\kappa$ be an infinite cardinal, and let $\mathbb{P}$ be a forcing notion of size less than $\kappa$. Let G be a $\mathbb{P}$-generic filter over V.
\begin{enumerate}
\item If $\kappa$ is Ramsey in V, then $\kappa$ is Ramsey in $V[G]$.
\item If $\kappa$ is measurable with a $\kappa$-complete ultrafilter $\mathcal{U}$ in V, then $\kappa$ is measurable with a $\kappa$-complete ultrafilter $\mathcal{U}_1=\{X\subseteq \kappa: X\in V[G],\text{and there is a } Y\in\mathcal{U} \text{ such that }Y\subseteq X\}$ defined in $V[G]$ generated by $\mathcal{U}$ in $V[G]$.
\end{enumerate}
}
\end{thm}
\begin{proof}
(1) follows from \cite[\textbf{Theorem 21.2}]{Jec2003} and (2) follows from \cite[\textbf{Theorem 10}]{LS1967}.
\end{proof}
\subsection{Symmetric extension} Symmetric extensions are submodels of the generic extension containing the ground model, where $\mathsf{AC}$ can consistently fail. Let $\mathbb{P}$ be a forcing notion, $\mathcal{G}$ be a group of automorphisms of $\mathbb{P}$ and $\mathcal{F}$ be a normal filter of subgroups over $\mathcal{G}$.
We recall the following Symmetry Lemma from \cite{Jec2003}.
\begin{lem}{\textbf{(The Symmetry Lemma; cf. \cite[\textbf{Lemma 14.37}]{Jec2003})}}
{\em Let $\mathbb{P}$ be a forcing notion, $\varphi$ be a formula of the forcing language with $n$ free variables and let $\sigma_1,\sigma_2,...,\sigma_n$ be $\mathbb{P}$-names. If $a$ is an automorphism of $\mathbb{P}$, then $p\Vdash \varphi(\sigma_1,\sigma_2,...,\sigma_n) \Longleftrightarrow a(p)\Vdash \varphi(a(\sigma_1),a(\sigma_2),...,a(\sigma_n))$.
}
\end{lem}
For $\tau\in V^{\mathbb{P}}$, we denote its symmetric group with respect to $\mathcal G$ by $sym^{\mathcal G}(\tau) =\{g\in \mathcal G : g\tau = \tau\}$ and say $\tau$ is {\em symmetric} with respect to $\mathcal{F}$ if $sym^{\mathcal G}(\tau)\in\mathcal F$. Let $HS^\mathcal F$ be the class of all hereditary symmetric names.
That is, recursively for $\tau\in V^{\mathbb{P}}$,
\begin{center}
$\tau\in HS^{\mathcal{F}}$ iff $\tau$ is symmetric with respect to $\mathcal{F}$, and for each $\sigma\in dom(\tau)$, $\sigma\in HS^{\mathcal{F}}$.
\end{center}
We define the symmetric extension of $V$ with respect to $\mathcal F$ as $V(G)^\mathcal F = \{\tau^G : \tau\in HS^\mathcal F\}$. For the sake of our convenience we omit the superscript $\mathcal{F}$ sometimes and call $V(G)^\mathcal F$ as $V(G)$, $HS^\mathcal F$ as $HS$, and $sym^\mathcal F(\tau)$ as $sym(\tau)$.
\begin{defn}{\textbf{(Symmetric system; cf. \cite[\textbf{Definition 2.1}]{HK2019})}} We say $\langle\mathbb{P},\mathcal{G},\mathcal{F}\rangle$ is a {\em symmetric system} if $\mathbb{P}$ is a forcing notion, $\mathcal{G}$ is a group of automorphisms of $\mathbb{P}$, and $\mathcal{F}$ is a normal filter of subgroups over $\mathcal{G}$.
\end{defn}
\begin{defn}{\textbf{(Tenacious system; \textbf{cf. \cite[\textbf{ Definition 4.6}]{Kar2019a}})}} Let $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$ be a symmetric system. A condition $p\in \mathbb{P}$ is {\em $\mathcal{F}$-tenacious} if $\{\pi\in\mathcal{G} : \pi(p)= p\}\in \mathcal{F}$. We say $\mathbb{P}$ is {\em $\mathcal{F}$-tenacious} if there is a dense subset of $\mathcal{F}$-tenacious conditions. We say $\langle\mathbb{P},\mathcal{G},\mathcal{F}\rangle$ is a {\em tenacious system} if $\mathbb{P}$ is $\mathcal{F}$-tenacious.
\end{defn}
Karagila and Hayut proved that every symmetric system is equivalent to a tenacious system (cf. \cite[\textbf{Appendix A}]{Kar2019a}). Thus, it is natural to assume tenacity and work with tenacious system. We recall the following theorem which states that the symmetric extension $V(G)$ is a transitive model of $\mathsf{ZF}$.
\begin{thm}
{\textbf{(cf. \cite[\textbf{Lemma 15.51}]{Jec2003})}} {\em If $\langle\mathbb{P},\mathcal{G},\mathcal{F}\rangle$ is a symmetric system and G is a $\mathbb{P}$-generic filter over $V$, then $V(G)$ is a transitive model of $\mathsf{ZF}$ and $V\subseteq V(G)\subseteq V[G]$.}
\end{thm}
\subsection{Terminologies} We recall the terminologies like {\em Approximation Lemma}, {\em Approximation property}, and {\em ($\mathcal{G},\mathcal{I}$)-homogeneous forcing notion}, from \cite{Dim2011} and \cite{Ina2013}.
For $E\subseteq\mathbb{P}$, let us define its pointwise stabilizer group to be
fix$_{\mathcal{G}}E = \{g\in\mathcal{G}:\forall p \in E (g(p)=p)\}$,
i.e., it is the set of automorphisms which fix $E$ pointwise. We denote fix$_{\mathcal{G}}E$ by fix $E$ for the sake of convenience.
\begin{defn}{\textbf{($\mathcal G$-symmetry generator; \cite[\textbf{Definition 14}]{Ina2013})}}
Let $\mathbb{P}$ be a forcing notion and $\mathcal{G}$ be a group of automorphisms of $\mathbb{P}$. A subset $\mathcal{I}\subseteq \mathcal P(\mathbb{P})$ is called a $\mathcal G$-{\em symmetry generator} if it consists of up-sets, is closed under unions, and if for all $g\in \mathcal G$ and $E\in \mathcal{I}$, there is an $E'\in\mathcal{I}$ such that $g$(fix$E$)$g^{-1}\supseteq$ fix$E'$.
\end{defn}
We can see that if $\mathcal{I}$ is a $\mathcal G$-symmetry generator, then the set $\{$fix$E: E\in\mathcal{I}\}$ generates a normal filter over $\mathcal G$ (cf. \cite[\textbf{Lemma 15}]{Ina2013}).
\begin{defn}
Let $\mathcal{I}$ be the $\mathcal G$-symmetry generator, we say $E\in \mathcal{I}$ {\em supports} a name $\sigma\in HS$ if fix$E\subseteq sym (\sigma$).
\end{defn}
\begin{defn}{\textbf{(Projectable $\mathcal{G}$-symmetry generator; \cite[\textbf{Definition 1.25}]{Dim2011} $\&$ \cite[\textbf{Definition 17}]{Ina2013})}}
Let $\mathbb{P}$ be a forcing notion, $\mathcal{G}$ be a group of automorphisms of $\mathbb{P}$, and $\mathcal{I}$ be a $\mathcal{G}$-symmetry generator.
We say $\mathcal{I}$ is {\em projectable} for the pair ($\mathbb{P},\mathcal{G}$) if for every $p\in\mathbb{P}$ and every $E\in \mathcal{I}$, there is a $p^{*}\in E$ that is minimal (with respect to the partial order) and unique such that $p^{*}\geq p$.
We call $p\restriction E = p^{*}$ the {\em projection} of $p$ to $E$.
\end{defn}
For the rest of this section, let $\mathbb{P}$ be a forcing notion, $\mathcal{G}$ be a group of automorphisms of $\mathbb{P}$, and $\mathcal{I}$ be a projectable $\mathcal{G}$-symmetry generator for the pair ($\mathbb{P},\mathcal{G}$).
\begin{defn}{\textbf{(Approximation property; \cite[\textbf{Definition 18}]{Ina2013})}}
We say that the triple $\langle\mathbb{P},\mathcal{G},\mathcal{I}\rangle$ has the {\em approximation property} if for any formula $\varphi$ with $n$ free variables, and names $ \sigma_1,\sigma_2,...,\sigma_n\in HS$ all with support $E\in\mathcal{I}$, and for any $p\in \mathbb{P}$, $p\Vdash\varphi(\sigma_1,\sigma_2,...,\sigma_n)$ implies that $p\restriction E \Vdash\varphi(\sigma_1,\sigma_2,...,\sigma_n) $.
\end{defn}
\begin{defn}{\textbf{($(\mathcal{G},\mathcal{I})$-homogeneous forcing notion; \cite[\textbf{Definition 1.26}]{Dim2011} $\&$ \cite[\textbf{Definition 19}]{Ina2013})}}
We say that $\mathbb{P}$ is {\em $(\mathcal{G},\mathcal{I})$-homogeneous} if for every $E\in \mathcal{I}$, every $p\in\mathbb{P}$, and every $q\in\mathbb{P}$ such that $q\leq p\restriction E$, there is an automorphism $a\in$ fix$E$ such that $a(p)\parallel q$.
\end{defn}
\begin{lem}
{\textbf{(\cite[\textbf{Lemma 1.27}]{Dim2011} $\&$ \cite[\textbf{Lemma 20}]{Ina2013})}}
{\em If $\mathbb{P}$ is $(\mathcal{G},\mathcal{I})$-homogeneous, then $\langle\mathbb{P},\mathcal G,\mathcal{I}\rangle$ has the approximation property.}
\end{lem}
We note that if $E \in \mathcal{I}$, then $E$ itself is a forcing notion (with the same top element as $\mathbb{P}$). So, we can say that set of pairs $\tau$ is an $E$-name iff $\tau$ is a relation and for every $(\sigma, p) \in\tau$, $\sigma$ is an $E$-name and $p \in E$.
\begin{lem}
{\textbf{(Approximation Lemma; \cite[\textbf{Lemma 1.29}]{Dim2011} $\&$ \cite[\textbf{Lemma 21}]{Ina2013})}}
{\em If the triple $\langle\mathbb{P},\mathcal G,\mathcal{I}\rangle$ has the approximation property then for all set of ordinals $X\in V(G)$, there exists an $E\in \mathcal{I}$ and an $E$-name for $X$. Thus, $X\in V[G\cap E]$.}
\end{lem}
\subsection{Homogeneity of forcing notions} We recall the definition of {\em weakly homogeneous} and
{\em cone homogeneous} forcing notion from \cite{DF2008}.
\begin{defn}{\textbf{(Weakly homogeneous forcing notion; \cite[\textbf{Definition 2}]{DF2008})}}
We say a set forcing notion $\mathbb{P}$ is {\em weakly homogeneous} if and only if for any $p,q\in \mathbb{P}$, there is an automorphism $a:\mathbb{P}\rightarrow\mathbb{P}$ such that $a(p)$ and $q$ are compatible.\footnote{The Levy collapse $Col(\lambda,<\kappa)$ is weakly homogeneous, given an infinite cardinal $\kappa$ and a regular cardinal $\lambda$.}
\end{defn}
\begin{defn}{\textbf{(Cone homogeneous forcing notion; \cite[\textbf{Definition 2}]{DF2008})}}
For $p\in\mathbb{P}$, let $Cone(p)$ denote $\{r\in\mathbb{P}:r\leq p\}$, the cone of conditions in $\mathbb{P}$ below $p$. We say a set forcing notion $\mathbb{P}$ is {\em cone homogeneous} if and only if for any $p, q\in \mathbb{P}$, there exist $p'\leq p$, $q'\leq q,$ and an isomorphism $\pi: Cone(p')\rightarrow Cone(q')$.
\end{defn}
If $\mathbb{P}$ is weakly homogeneous, then it is cone homogeneous too (cf. \cite[\textbf{Fact 1}]{DF2008}). Also, the finite support products of weakly (cone) homogeneous forcing notions are weakly (cone) homogeneous. A crucial feature of symmetric extensions using weakly (cone) homogeneous forcings are that they can be approximated by certain intermediate submodel where $\mathsf{AC}$ holds.
\subsection{Failure of weak choice principles}
We use $\mathsf{AC_{\kappa}}$ to denote the statement ``Every family of $\kappa$ non-empty sets admits a choice function''. We note that if $\kappa^{+}$ is singular, then $\mathsf{AC_{\kappa}}$ fails. This is due to the following well known fact.
\begin{fact}
{\em For all successor cardinal $\lambda$, $\mathsf{AC_{\kappa}}$ implies $cf(\lambda)>\kappa$.}
\end{fact}
We sketch another way of refuting $\mathsf{AC_{\kappa}}$.
For sets $A$ and $B$, we use $\mathsf{AC_{A}(B)}$ to denote the statement ``for each set $X$ of non-empty subsets of $B$, if there is an injection from $X$ to $A$ then there is a choice function for $X$". We recall \cite[\textbf{Lemmas 0.2, 0.3, 0.12}]{Dim2011}.
\begin{itemize}
\item Under $\mathsf{AC_{A}(B)}$, if there is a surjection from $B$ to $A$, then there is an injection from $A$ to $B$ (cf. \cite[\textbf{Lemma 0.2}]{Dim2011}).
\item For every infinite cardinal $\kappa$, there is a surjection from $\mathcal{P}(\kappa)$ onto $\kappa^{+}$ in $\mathsf{ZF}$ (cf. \cite[\textbf{Lemma 0.3}]{Dim2011}).
\item If $\kappa$ is measurable with a normal measure or $\kappa$ is weakly compact and $\alpha<\kappa$, then there is no injection $f:\kappa\rightarrow \mathcal{P}(\alpha)$ in $\mathsf{ZF}$ (cf. \cite[\textbf{Proposition 0.1}]{Bul1978}, \cite[\textbf{Lemma 0.12}]{Dim2011}).
\end{itemize}
The following lemma states that if a successor cardinal $\kappa$ is either measurable with a normal measure or weakly compact then $\mathsf{AC_{\kappa}}$ fails, which is \cite[\textbf{Corollary 0.3}]{Bul1978}.
\begin{lem}
{\em Let $\kappa=\alpha^{+}$ be a successor cardinal. If $\kappa$ is measurable with a normal measure or weakly compact then $\mathsf{AC_{\alpha^{+}}(\mathcal{P}(\alpha))}$ fails}.
\end{lem}
\begin{proof}
Let $\mathsf{AC_{\alpha^{+}}(\mathcal{P}(\alpha))}$ holds. We show $\kappa=\alpha^{+}$ is neither measurable with a normal measure nor weakly compact. In $\mathsf{ZF}$, there is a surjection from $\mathcal{P}(\alpha)$ onto $\alpha^{+}$. Now $\mathsf{AC_{\alpha^{+}}(\mathcal{P}(\alpha))}$ implies there is an injection $f'$ from $\alpha^{+}$ to $\mathcal{P}(\alpha)$ which states that $\kappa=\alpha^{+}$ is neither measurable with a normal measure nor weakly compact.
\end{proof}
\section{Preserving Dependent Choice in symmetric extensions}
Dependent Choice, denoted by $\mathsf{DC}$ or $\mathsf{DC_{\omega}}$, is a weaker version of $\mathsf{AC}$ which is strictly stronger\footnote{In Howard--Rubin's first model ($\mathcal{N}_{38}$ in \cite{HR1998}), $\mathsf{AC_{\omega}}$ holds but $\mathsf{DC_{\omega}}$ fails.} than the Countable Choice, denoted by $\mathsf{AC_{\omega}}$. This principle is strong enough to give the basis of analysis as it is equivalent to the Baire Category Theorem which is a fundamental theorem in functional analysis. Further, $\mathsf{DC}$ is equivalent to other important theorems like the countable version of the Downward L\"{o}weinheim--Skolem theorem and every tree of height $\omega$ without a maximum node has an infinite branch etc. On the other hand, $\mathsf{AC}$ has several controversial applications like the existence of a non-Lebesgue measurable set of real numbers, Banach--Tarski Paradox and the existence of a well-ordering of real numbers whereas $\mathsf{DC}$ does not have such counter-intuitive consequences. Thus it is desirable to preserve $\mathsf{DC}$ in symmetric extensions.
For an infinite cardinal $\kappa$, we denote by $\mathsf{DC_{\kappa}}$ the principle of Dependent Choice for $\kappa$. This principle states that for every non-empty set $X$, if $R$ is a binary relation such that for each ordinal $\alpha<\kappa$, and each $f:\alpha\rightarrow X$ there is some $y\in X$ such that $f$ $R$ $y$, then there is $f:\kappa\rightarrow X$ such that for each $\alpha<\kappa$, $f\restriction\alpha$ $R$ $f(\alpha)$.
We denote the assertion $(\forall\lambda<\kappa)\mathsf{DC_{\lambda}}$ by $\mathsf{DC_{<\kappa}}$. We recall that $\mathsf{AC}$ is equivalent to $\forall\kappa(\mathsf{DC_{\kappa}})$ and $\mathsf{DC_{\kappa}}$ implies $\mathsf{AC_{\kappa}}$. We refer the reader to \cite[\textbf{Chapter 8}]{Jec1973}, for details concerning $\mathsf{DC_{\kappa}}$ and related choice principles.
We recall the definitions of a forcing notion with the {\em $\kappa$-chain condition} ($\kappa$-c.c.) and
of forcing notions that are {\em $\kappa$-closed} and {\em $\kappa$-distributive} from \cite[\textbf{Definition 5.8}]{Cum2010}. We also recall the definitions of forcing notions that are {\em $<\kappa$-strategically closed}, {\em $\kappa$-strategically closed}, and {\em $(\kappa+1)$-strategically closed} from \cite[\textbf{Definition 5.14, Definition 5.15}]{Cum2010}.
Monro \cite[\textbf{Corollary 5.11}]{Mon1983} proved that $\mathsf{DC_{\kappa}}$ is
not preserved by generic extensions for any infinite cardinal $\kappa$.
Karagila \cite[\textbf{Lemma 1}]{Kar2014} proved that if $\mathbb{P}$ is $\kappa$-closed and $\mathcal{F}$ is $\kappa$-complete then $\mathsf{DC_{<\kappa}}$ is preserved in the symmetric extension in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$.
The author and Karagila both observed independently that ``$\mathbb{P}$ is $\kappa$-closed'' can be replaced by ``$\mathbb{P}$ has the $\kappa$-c.c.'' in \cite[\textbf{Lemma 1}]{Kar2014}. The author noticed this observation combining the role of $\kappa$-c.c. forcing notions from \cite[\textbf{Lemma 2.2}]{Apt2001} and the role of $\kappa$-completeness of $\mathcal{F}$ from \cite[\textbf{Lemma 1}]{Kar2014}.
The idea was the following. If $\mathbb{P}$ has the $\kappa$-c.c., then any antichain is of size less than $\kappa$. So by Zorn's Lemma in the ground model, there is a maximal antichain of conditions $\mathcal{A}=\{p_{\alpha}: \alpha<\gamma<\kappa\}$ extending $p$ such that for all $\alpha<\gamma$, $p_{\alpha}\Vdash \dot{f}(\hat{\alpha})=\dot{t}_{\alpha}$ where $\dot{t}_{\alpha}\in HS$. Then we can follow the proof of \cite[\textbf{Lemma 1}]{Kar2014} to finish the proof.
In a private conversation with Karagila, the author came to know that they independently observed the same fact. We note that there was a gap in the above observation. Specifically, the author was not aware of the fact that every symmetric system is equivalent to a tenacious system. Karagila fixed this gap. In particular, in \cite[\textbf{Lemma 3.3}]{Kar2019}, Karagila wrote that the natural assumption that $\langle\mathbb{P},\mathcal{G},\mathcal{F}\rangle$ is a tenacious system is also required in the proof.
\begin{lem}{(\textbf{Karagila}; \cite[\textbf{Lemma 3.3}]{Kar2019})}
{\em Let $V$ be a model of $\mathsf{ZFC}$. If $\mathbb{P}$ has the $\kappa$-c.c. and $\mathcal{F}$ is $\kappa$-complete, then $\mathsf{DC_{<\kappa}}$ is preserved in the symmetric extension of $V$ with respect to the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$.}
\end{lem}
We slightly generalize \cite[\textbf{Lemma 1}]{Kar2014} and prove \textbf{Observation 1.1}.
\begin{lem}{\textbf{(Observation 1.1)}}
{\em Let $V$ be a model of $\mathsf{ZFC}$. If $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete, then $\mathsf{DC_{<\kappa}}$ is preserved in the symmetric extension of $V$ with respect to the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$.}
\end{lem}
\begin{proof}
Let $G$ be a $\mathbb{P}$-generic filter over $V$. Let $\delta<\kappa$. We show that $\mathsf{DC_{\delta}}$ holds in $V(G)$. Let $X$ and $R$ be elements of $V(G)$ as in the assumptions of $\mathsf{DC_{\delta}}$. Since $\mathsf{AC}$ is equivalent to $\forall\kappa$($\mathsf{DC_{\kappa}}$) and $V[G]$ is a model of $\mathsf{AC}$, using $\forall\kappa$($\mathsf{DC_{\kappa}}$) in $V[G]$, we can find an $f:\delta\rightarrow X$ in $V[G]$ such that $f\restriction \alpha$ $R$ $f(\alpha)$ for all $\alpha < \delta$. We show that this $f$ is in $V(G)$.
Let $p_{0}\Vdash \dot{f}$ is a function whose domain is $\delta$ and range is $X$ which is a subset of $V(G)$.
For each $\alpha<\delta$, $D_{\alpha}=\{p\leq p_{0}: (\exists x\in X) p\Vdash \dot{f}(\check{\alpha})=\dot{x}$ where $\dot{x}\in HS \}$ is open dense below $p_{0}$. Consequently by $\kappa$-distributivity of $\mathbb{P}$, $D=\bigcap_{\alpha<\delta} D_{\alpha}$ is dense below $p_{0}$. So, there is some $p\in D\cap G$. We can see that for each $\alpha<\delta$, there is an $x_{\alpha}$ such that $p\Vdash \dot{f}(\check{\alpha})=\dot{x}_{\alpha}$ where $\dot{x}_{\alpha}\in HS$. Define $\dot{g}=\{\langle\check{\alpha},\dot{x}_{\alpha}\rangle:\alpha<\delta\}$. Now, since each $\dot{x}_{\alpha}\in HS$, $sym(\dot{x}_{\alpha})\in \mathcal{F}$. By $\kappa$-completeness of $\mathcal{F}$,
$H=\bigcap_{\alpha<\delta}sym(\dot{x}_{\alpha})\in \mathcal{F}$. Next, since $H$ is a subgroup of $sym{(\dot{g})}$
and $\mathcal{F}$ is a filter, $\dot{g}\in HS$. We can see that $p\Vdash \dot{g}=\dot{f}$. Thus, there is a dense open set of conditions $q\leq p_{0}$, such that for some $\dot{g}\in HS$, $q\Vdash\dot{g}=\dot{f}$. By genericity, $\dot{f}^{G}=f\in V(G)$.
\end{proof}
\begin{remark}If $\kappa$ is either a supercompact cardinal or a strongly compact cardinal and $\lambda>\kappa$ is a regular cardinal, there are certain forcing notions like supercompact Prikry forcing \cite{Apt1985} and strongly compact Prikry forcing \cite{AH1991} which are known to be non-$\kappa$-distributive, but still can preserve $\mathsf{DC_{\kappa}}$ in the symmetric extension based on such forcings.
In particular, Apter communicated to us that, assuming the consistency of a $2^{\lambda}$-supercompact cardinal $\kappa$ and a regular cardinal $\lambda>\kappa$, Kofkoulis proved in \cite{Kof1990}, that in a symmetric extension based on supercompact Prikry forcing, $\mathsf{DC_{\kappa}}$ was preserved. In particular, $\mathsf{DC_{\kappa}}$ holds in the symmetric inner model constructed in \cite[\textbf{Theorem 1}]{Apt1985}. Further applying the methods of Kofkoulis, assuming the consistency of a $\lambda$-strongly compact cardinal $\kappa$ and a measurable cardinal $\lambda>\kappa$, a symmetric extension based on strongly compact Prikry forcing was constructed in \cite{AH1991} where $\kappa$ became a singular cardinal of cofinality $\omega$, $\kappa^{+}$ remained a measurable cardinal and $\mathsf{DC_{\kappa}}$ was preserved. We can also find another exhibition of Kofkoulis's method with certain modifications in \cite{AM1995}.
\end{remark}
Next, we prove \textbf{Observation 1.2}.
\begin{lem}{\textbf{(Observation 1.2)}}
{\em Let $\delta<\kappa$ and $V$ be a model of $\mathsf{ZF+DC_{\delta}}$. If $\mathbb{P}$ is ($\delta+1$)-strategically closed and $\mathcal{F}$ is $\kappa$-complete, then $\mathsf{DC_{\delta}}$ is preserved in the symmetric extension of $V$ with respect to the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$. }
\end{lem}
\begin{proof}
Let $\delta<\kappa$. Let $G$ be a $\mathbb{P}$-generic filter over $V$. By \cite[\textbf{Theorem 2.2}]{GJ2014}, $\mathsf{DC_{\delta}}$ is preserved in $V[G]$ since $\mathbb{P}$ is ($\delta+1$)-strategically closed. We show that $\mathsf{DC_{\delta}}$ holds in $V(G)$. Let $X$ and $R$ be elements of $V(G)$ as in the assumptions of $\mathsf{DC_{\delta}}$. Since $\mathsf{DC_{\delta}}$ is preserved in $V[G]$, we can find an $f:\delta\rightarrow X$ in $V[G]$ such that $f\restriction \alpha$ $R$ $f(\alpha)$ for all $\alpha < \delta$. We show that this $f$ is in $V(G)$.
Let $p\Vdash \dot{f}$ is a function whose domain is $\delta$ and its range is a subset of $V(G)$.
Consider a game of length $\delta+1$, between two players I and II who play at odd stages and even stages respectively such that initially II chooses a trivial condition and I chooses a condition $p_{0}$ extending $p$ such that $p_{0}\Vdash \dot{f}(\check{0})=\dot{t}_{0}$ where $\dot{t}_{0}$ is in $HS$, and at non-limit even stages 2$\alpha>0$, II chooses a condition $p_{\alpha}$ extending the condition of the previous stage such that $p_{\alpha}\Vdash \dot{f}(\check{\alpha})=\dot{t}_{\alpha}$ where $\dot{t}_{\alpha}$ is in $HS$. By ($\delta+1$)-strategic closure of $\mathbb{P}$, II has winning strategy. Thus, we can we can extend
$p$ to $p_{0} \geq p_{1} \geq$ ··· $\geq p_{\alpha}$ $\geq$ ···$\geq p_{\delta}$ such that $p_{\alpha}\Vdash \dot{f}(\check{\alpha})=\dot{t}_{\alpha}$ where $\dot{t}_{\alpha}$ is in $HS$ for each $\alpha<\delta$. It is enough to show that $\dot{f}=\{\langle \check{\beta},\dot{t}_{\beta}\rangle: \beta<\delta\}$ is in $HS$ which follows using $\kappa$-completeness of $\mathcal{F}$ as done in \cite[\textbf{Lemma 1}]{Kar2014}.
\end{proof}
\begin{remark}
We note that we are using the definition of a $(\delta+1)$-strategically closed forcing notion
from \cite[\textbf{Definition 5.15}]{Cum2010}
which is different from the definition used in \cite{GJ2014}. In particular, in our case a forcing notion is $\delta$-strategically closed if in the two person game in which the players construct a descending sequence $\langle p_{\alpha} : \alpha < \delta\rangle$, where player I plays odd stages and player II plays even and limit stages
(choosing the trivial condition at stage 0), player II has a strategy which ensures the game
can always be continued; a forcing notion is ($\delta+1$)-strategically closed if the corresponding game has length $\delta+1$.
Whereas Gitman and Johnstone defined that a forcing notion is $\leq\delta$-strategically closed if in the game of ordinal length $\delta+1$ in which two players alternatively select conditions from it to construct a descending
$\delta+1$-sequence with the second player playing at even and limit stages, the second player has a
strategy that allows her to always continue playing (cf. \cite[the paragraph before \textbf{Theorem 2.2}]{GJ2014}).
Thus in our case if $V$ is a model of $\mathsf{ZF+DC_{\delta}}$ and $\mathbb{P}$ is ($\delta+1$)-strategically closed then $\mathsf{DC_{\delta}}$ is preserved in $V[G]$ by \cite[\textbf{Theorem 2.2}]{GJ2014}.
\end{remark}
\begin{remark}
Let $V$ be a model of $\mathsf{ZF +DC_{\kappa}}$. Suppose $\mathbb{P}$ is well-orderable of order type at most $\kappa$ and has the $\kappa$-c.c. property. We remark that if $\mathcal{F}$ is $\kappa$-complete, then $\mathsf{DC_{<\kappa}}$ is preserved in the symmetric extension of $V$ in terms of the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$.
Let $G$ be a $\mathbb{P}$-generic filter over $V$.
By \cite[\textbf{Theorem 2.1}]{GJ2014}, $\mathsf{DC_{\kappa}}$ is preserved in $V[G]$. The rest follows from the proof of \cite[\textbf{Lemma 3.3}]{Kar2019}.
\end{remark}
\begin{question}
{\em Let V be a model of $\mathsf{ZF +DC_{\kappa}}$. Suppose that $\mathbb{P}$ is $\kappa$-distributive. Can we preserve $\mathsf{DC_{\kappa}}$ in every forcing extension $V[G]$ by $\mathbb{P}$?}
\end{question}
If the answer is in the affirmative, then we can say that if $V$ is a model of $\mathsf{ZF +DC_{\kappa}}$, $\mathbb{P}$ is $\kappa$-distributive, and $\mathcal{F}$ is $\kappa$-complete, then $\mathsf{DC_{<\kappa}}$ is preserved in the symmetric extension in terms of the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$ following \textbf{Lemma 3.2}.
\subsection{Number of normal measures a successor cardinal can carry and Dependent Choice}
Takeuti \cite{Tak1970} and Jech \cite{Jec1968} independently proved that
if we assume the consistency of ``$\mathsf{ZFC}$ + there is a measurable cardinal" then the theory ``$\mathsf{ZF+DC}$ + $\aleph_{1}$ is a measurable cardinal'' is consistent.
Dimitriou \cite[\textbf{section 1.33}]{Dim2011} modified Jech's construction and proved that if we assume the consistency of ``$\mathsf{ZFC}$ + there is a measurable cardinal $\kappa$ and $\gamma<\kappa$ is a regular cardinal'' then the theory ``$\mathsf{ZF}$ + the cardinality of $\gamma$ is preserved + $\gamma^{+}$ is a measurable cardinal'' is consistent. Apter, Dimitriou, and Koepke \cite{ADK2014} constructed symmetric models in which for an arbitrary ordinal $\rho$, $\aleph_{\rho+1}$ can be the least measurable as well as the least regular uncountable cardinal.
Bilinsky and Gitik \cite{BG2012} proved that if we assume the consistency of ``$\mathsf{ZFC+GCH}$ + there is a measurable cardinal $\kappa$'' then we can obtain a symmetric extension where $\kappa$ is a measurable cardinal without a normal measure.
Assuming the consistency of {``$\mathsf{ZFC+GCH}$ + there is a measurable cardinal"}, we can construct models of $\mathsf{ZF+DC}$ where successor of regular cardinals like $\aleph_1$, $\aleph_{2}$, $\aleph_{\omega+2}$, as well as $\aleph_{\omega_{1}+2}$, can carry an arbitrary (non-zero) number of normal measures.
Friedman--Magidor \cite[\textbf{Theorem 1}]{FM2009} proved that a measurable cardinal can be forced to carry arbitrary number of normal measures in $\mathsf{ZFC}$.
\begin{lem}{(\textbf{Friedman and Magidor}; \cite[\textbf{Theorem 1}]{FM2009})} {\em Assume $\mathsf{GCH}$. Suppose that $\kappa$ is a measurable cardinal and let $\alpha$ be a cardinal at most $\kappa^{++}$. Then in a cofinality-preserving forcing extension, $\kappa$ carries exactly $\alpha$ normal measures.}\end{lem}
We recall the definition of a {\em symmetric collapse} from \cite{HK2019}.
\begin{defn}{(\textbf{Symmetric Collapse}; \cite[\textbf{Definition 4.1}]{HK2019})} Let $\kappa\leq\lambda$ be two infinite cardinals. The {\em symmetric collapse} is the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$ defined as follows.
\begin{itemize}
\item $\mathbb{P}=Col(\kappa,<\lambda)$, so a condition in $\mathbb{P}$ is a partial function $p$ with domain
$\{\langle \alpha, \beta\rangle : \kappa < \alpha < \lambda, \beta < \kappa\}$ such that $p(\alpha,\beta) < \alpha$ for all $\alpha$ and $\beta$,
supp($p$) = $\{\alpha < \lambda : \exists\beta, \langle \alpha, \beta\rangle \in$ dom $p\}$ is bounded below $\lambda$ and $\vert p\vert < \kappa$.
\item $\mathcal{G}$ is the group of automorphisms $\pi$ such that there is a sequence of permutations $\overrightarrow{\pi}=\langle\pi_{\alpha}:\kappa<\alpha<\lambda \rangle$ such that $\pi_{\alpha}$ is a permutation of $\alpha$ satisfying $\pi p(\alpha,\beta)=\pi_{\alpha}(p(\alpha,\beta))$.
\item $\mathcal{F}$ is the normal filter of subgroups generated by fix$(E)$ for bounded $E\subseteq\lambda$, where fix$(E)$ is the group $\{\pi: \forall\alpha\in E, \pi p(\alpha,\beta)=p(\alpha,\beta)\}$.\end{itemize}
\end{defn}
\begin{lem}{\em Let $\kappa\leq\lambda$ be two infinite cardinals such that $cf(\lambda)\geq\kappa$ and $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$ is the symmetric collapse where $\mathbb{P}=Col(\kappa,<\lambda)$. Then, $\mathcal{F}$ is $\kappa$-complete.}\end{lem}
\begin{proof}Fix $\gamma<\kappa$ and let, for each $\beta<\gamma$, $K_{\beta}\in \mathcal{F}$. There must be bounded $E_{\beta}\subseteq\lambda$ for each $\beta<\gamma$ such that fix$E_{\beta}\subseteq K_{\beta}$. Next, fix($\bigcup_{\beta<\gamma}E_{\beta})\subseteq\bigcap_{\beta<\gamma}$ fix$E_{\beta}\subseteq \bigcap_{\beta<\gamma} K_{\beta}$. Since $cf(\lambda)\geq\kappa$, $\bigcup_{\beta<\gamma}E_{\beta}$ is a bounded subset of $\lambda$. Consequently, $\bigcap_{\beta<\gamma} K_{\beta}\in \mathcal{F}$.\end{proof}
We observe that after a symmetric collapse, the successor of a regular cardinal can be a measurable cardinal carrying an arbitrary (non-zero) number of normal measures assuming the consistency of a measurable cardinal. Further we can preserve Dependent Choice in certain cases.
\begin{thm}
{\em Let $V$ be a model of $\mathsf{ZFC+GCH}$ with a measurable cardinal $\kappa$. Let $\lambda$ be any non-zero cardinal at most $\kappa^{++}$ and let $\eta\leq\kappa$ be regular. Then, there is a symmetric extension where $\kappa=\eta^{+}$ is a measurable cardinal carrying $\lambda$ normal measures. Moreover, $\mathsf{AC_{\kappa}}$ fails and $\mathsf{DC_{<\eta}}$ holds\footnote{If we assume $\eta>\omega$.} in the symmetric model}.\end{thm}
\begin{proof} Applying \textbf{Lemma 3.8}, we obtain a cofinality-preserving forcing extension $V'$ of $V$ where $\kappa$ is a measurable cardinal with $\lambda$ many normal measures. Let $V'(G)$ be the symmetric extension of $V'$ obtained by the symmetric collapse $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$ where $\mathbb{P}=Col(\eta,<\kappa)$ and $G$ is a $\mathbb{P}$-generic filter over $V'$. In $V'(G)$, $\kappa=\eta^{+}$. We can also have the following in $V'(G)$.
\begin{itemize}
\item By \cite[\textbf{Lemmas 2.4, 2.5}]{Apt2001}, $\kappa$ remains a measurable cardinal with $\lambda$ many normal measures. \item Since $\kappa$ is a successor as well as a measurable cardinal, $\mathsf{AC_{\kappa}}$ fails using \textbf{Lemma 2.20}.
\item We note that $\mathcal{F}$ is $\eta$-complete by \textbf{Lemma 3.10}.
Since $\mathbb{P}$ is $\eta$-closed, $\mathsf{DC_{<\eta}}$ holds using \cite[\textbf{Lemma 1}]{Kar2014}. \end{itemize}\end{proof}
\begin{remark}
The referee pointed out that $\mathsf{DC_{<\kappa}}$ is preserved in $V'(G)$. Assuming that $\lambda$ is regular, the proof of \textbf{Lemma 3.10} gives that $\mathcal{F}$ is $\lambda$-complete. Consequently, since $\kappa$ is a regular cardinal in $V'$, $\mathcal{F}$ is $\kappa$-complete. Since $\mathbb{P}$ has the $\kappa$-c.c., by \textbf{Lemma 3.1}, $\mathsf{DC_{<\kappa}}$ is preserved in $V'(G)$.
\end{remark}
\begin{remark}
In \cite[\textbf{Theorem 1}]{Apt2001},
starting with a model of $``\mathsf{ZFC}+ \mathsf{GCH}+ o(\kappa)=\delta^{*}"$ for $\delta^{*}\leq\kappa^{+}$ any finite or infinite cardinal, Apter constructed a model of $\mathsf{ZF}+\mathsf{DC_{<\kappa}}$ where $\kappa$ carries exactly $\delta^{*}$ normal measures and $2^{\delta} = \delta^{++}$ on a set having measure one with respect to every
normal measure over $\kappa$. We observe that we can obtain the result of \cite[\textbf{Theorem 1}]{Apt2001} starting from just one measurable cardinal $\kappa$ if we use \textbf{Lemma 3.8} instead of passing to an inner model of Mitchell from \cite{Mit1974} as done in the proof of \cite[\textbf{Theorem 1}]{Apt2001}. In particular, we can prove the following.
\end{remark}
\begin{corr}{\textbf{(of \cite[\textbf{Theorem 1}]{Apt2001})}}
{\em Let $V$ be a model of $\mathsf{ZFC + GCH}$ with a measurable cardinal $\kappa$ and let $\lambda$ be a cardinal at most $\kappa^{++}$. Then there is a model of $\mathsf{ZF+\mathsf{DC_{<\kappa}}}$ where $\kappa$ is a measurable cardinal carrying $\lambda$ many normal measures $\langle \mathcal{U}_{\alpha}^{*}: \alpha<\lambda\rangle$. Moreover, we have $2^{\delta} = \delta^{++}$ on a set having measure one with respect to any of
the measures $\mathcal{U}_{\alpha}^{*}$.}
\end{corr}
\section{The Proof of Theorem 1.5}
In this section we prove \textbf{Theorem 1.5}.
\begin{proof}{\textbf{(of Theorem 1.5)}}
Firstly, we give a description of the symmetric extension constructed in \cite[\textbf{Theorem 4.1}]{Kar2019} as follows.
\begin{enumerate}
\item \textbf{The ground model ($V$).}
At the beginning of the proof of \cite[\textbf{Theorem 3}]{Apt2012}, from the given requirements, Apter constructed a model $V$ where
there is an enumeration $\langle\kappa_{i} : i <\kappa\rangle$ of $C\cup \{\omega\}$ where $C\subseteq\kappa$ is a club of inaccessible and limit cardinals below a supercompact cardinal $\kappa$ such that $2^{\kappa_{i}} = \kappa_{i}^{++}$ holds. We consider $V$ to be our ground model. For reader's convenience we recall the steps from the proof of \cite[\textbf{Theorem 3}]{Apt2012} as follows.
\begin{itemize}
\item Let $V$ be a model of $\mathsf{ZFC + GCH}$ with a supercompact cardinal $\kappa$.
\item Let $\mathbb{Q}_{1}$ be Laver’s partial ordering of \cite{Lav1978} which makes $\kappa$’s supercompactness indestructible under $\kappa$-directed closed forcing. Since $\mathbb{Q}_{1}$ may be defined so that $\vert\mathbb{Q}_{1}\vert=\kappa$, we have $V^{\mathbb{Q}_{1}* \dot{Add}(\kappa,\kappa^{++})}$= $V_{2}$ is a model of ``$\mathsf{ZFC}$ + $\kappa$ is supercompact + $2^{\kappa} = \kappa^{++}$ + $2^{\delta} = \delta^{+}$ for every cardinal $\delta \geq \kappa^{+}$".
\item Let $\mathbb{Q}_{3}\in V_{2}$ be the Radin forcing defined over $\kappa$. Taking a suitable measure sequence will enable one to preserve the supercompactness of $\kappa$ (cf. \cite{Git2010}). Consequently, $V_{2}^{\mathbb{Q}_{3}}= \overline{V}$ is a model of ``$\mathsf{ZFC}$ + $\kappa$ is supercompact + $2^{\kappa} = \kappa^{++}$ + $2^{\delta} = \delta^{+}$ for every cardinal $\delta \geq \kappa^{+}$ + There is a club $C \subseteq \kappa$ composed of inaccessible cardinals and their limits with $2^{\delta} = 2^{\delta^{+}}=\delta^{++}$ for every $\delta\in C$".
\item For the sake of convenience we consider the ground model to be $\overline{V} = V$. Let $\langle \kappa_{i}: i<\kappa\rangle\in V$ be the continuous,
increasing enumeration of $C\cup \{\omega\}$.
\end{itemize}
\item \textbf{Defining the symmetric system $\langle\mathbb{P},\mathcal{G},\mathcal{F}\rangle$.}
\begin{itemize}
\item Let $\mathbb{P}$ be the Easton support product of $\mathbb{P}_{\alpha}=Col(\kappa_{\alpha}^{++}, <\kappa_{\alpha+1})$ where $\alpha<\kappa$.
\item Let $\mathcal{G}$ be the Easton support product of the automorphism groups of each $\mathbb{P}_{\alpha}$.
\item Let $\mathcal{F}$ be the filter generated by the groups of the form fix$(\alpha)$ for $\alpha<\kappa$, where
fix$(\alpha)=\{\pi\in \prod _{\beta<\kappa}Aut(\mathbb{P}_{\beta}): \pi\restriction\alpha = id\}$.
\end{itemize}
\item \textbf{Defining the symmetric extension of $V$.} Let $G$ be a $\mathbb{P}$-generic filter over $V$. We consider the symmetric extension $V(G)^{\mathcal{F}}$ with respect to the symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$ defined above in (2) and denote it by $\mathcal{N}_{1}$ for the sake of our convenience.
\end{enumerate}
Since $\mathcal{F}$ is $\kappa$-complete, and $\mathbb{P}$ has the $\kappa$-c.c., $\mathsf{DC_{<\kappa}}$ is preserved in $\mathcal{N}_{1}$ by \textbf{Lemma 3.1} (cf. \cite[\textbf{section 4.1}]{Kar2019}). Since each $\mathbb{P}_{\alpha}$ is weakly homogeneous, the following lemma holds as a corollary of \textbf{Lemma 2.16}.
\begin{lem}
{\em If $A\in \mathcal{N}_{1}$ is a set of ordinals, then $A\in V[G\restriction\alpha]$ for some $\alpha<\kappa$.}
\end{lem}
We apply \textbf{Lemma 4.1} to prove that $\kappa$ remains supercompact in $\mathcal{N}_{1}$ following the methods of \cite{Ina2013}. Inamder \cite{Ina2013} proved that if we assume the consistency of ``$\mathsf{ZFC}$ + there is a supercompact cardinal $\kappa$, and $\gamma<\kappa$ is a regular cardinal'' then the theory ``$\mathsf{ZF}$ + the cardinality of $\gamma$ is preserved + $\gamma^{+}$ is a supercompact cardinal'' is consistent.
We recall the relevant lemmas and incorporate the arguments from \cite{Ina2013} in order to show that $\kappa$ remains supercompact in $\mathcal{N}_{1}$.
\begin{lem}{(cf. \cite[\textbf{Lemma 26}]{Ina2013})} {\em Let $\kappa$ be a regular cardinal, let $\gamma\geq \kappa$, and let $\mathbb{P}$ be a forcing notion of size less than $\kappa$. Then for every $C\in \mathcal{P}_{\kappa}(\gamma)^{V[G]}$, there is a $D\in \mathcal{P}_{\kappa}(\gamma)^{V}$ such that in $V[G]$, $C\subseteq D$.}
\end{lem}
\begin{lem}{(\textbf{L\'{e}vy--Solovay Lemma; \cite[\textbf{Lemma 27}]{Ina2013}})} {\em In $V$, let $\kappa$ be a regular cardinal, $D$ be a set and $\mathcal{U}$ be a $\kappa$-complete ultrafilter on $D$. Let $\mathbb{P}$ be a forcing notion of size less than $\kappa$ and $G$ be a $\mathbb{P}$-generic filter over $V$. Suppose $V[G]\models f:D\rightarrow V$. Then there is an $S\in\mathcal{U}$ and a $g:S\rightarrow V$ in $V$ such that $V[G]\models f\restriction S=g$.}
\end{lem}
Applying \textbf{Lemma 4.1} and \textbf{Lemma 4.3} we obtain the following lemma, which is similar to \cite[\textbf{Lemma 33}]{Ina2013}.
\begin{lem}
{\em Let $D$ be a set and $\mathcal{U}$ be a $\kappa$-complete ultrafilter on $D$ in $V$. Suppose $\mathcal{N}_{1}\models f:D\rightarrow V$. Then there is an $S\in \mathcal{U}$ and a $g:S\rightarrow V$ in $V$ such that $\mathcal{N}_{1}\models f\restriction S=g$.}
\end{lem}
\begin{proof}
By \textbf{Lemma 4.1}, for some $\alpha<\kappa$ we get $f\in V[G \restriction \alpha]$. Now we can say $G \restriction\alpha$ is $\mathbb{P}'$-generic over $V$ where $\vert\mathbb{P}'\vert<\kappa$. By \textbf{Lemma 4.3}, we obtain an $S\in\mathcal{U}$ and a $g:S\rightarrow V$ in $V$ such that $V[G\restriction\alpha]\models f\restriction S=g$. So, $\mathcal{N}_{1}\models f\restriction S=g$.
\end{proof}
Similarly \cite[\textbf{Lemma 34}]{Ina2013}, we obtain the following lemma by applying \textbf{Lemma 4.4}.
\begin{lem}
{\em In $V$, let $D$ be a set and $\mathcal{U}$ be a $\kappa$-complete ultrafilter on $D$. Let $\mathcal{W}$ be the filter on $D$ generated by $\mathcal{U}$ in $\mathcal{N}_{1}$. Then $\mathcal{W}$ is a $\kappa$-complete ultrafilter.}
\end{lem}
We follow the proof of \cite[\textbf{Theorem 35}]{Ina2013} and refer the reader to \cite{Ina2013} for further details.
\begin{lem}
{\em In $\mathcal{N}_{1}$, $\kappa$ is supercompact.}
\end{lem}
\begin{proof}
Let $\gamma\geq\kappa$ be arbitrary.
Since $\kappa$ is supercompact in $V$, there is a normal measure $\mathcal{U}$ on $\mathcal{P}_{\kappa}(\gamma)$ in $V$. Let $\mathcal{V}$ be the $\kappa$-complete measure it generates on $\mathcal{P}_{\kappa}(\gamma)^{V}$ in $\mathcal{N}_{1}$. Let $\mathcal{W}$ be the filter generated by $\mathcal{V}$ on $\mathcal{P}_{\kappa}(\gamma)$ in $\mathcal{N}_{1}$. Since $\mathcal{W}$ is generated by a $\kappa$-complete ultrafilter on $\mathcal{P}_{\kappa}(\gamma)^{V}\subseteq \mathcal{P}_{\kappa}(\gamma)$, $\mathcal{W}$ is a $\kappa$-complete ultrafilter by \textbf{Lemma 4.5}.
\textbf{Fineness}: Let $X\in \mathcal{P}_{\kappa} (\gamma)^ {\mathcal{N}_{1}}$. By \textbf{Lemma 4.1}, for some $\alpha<\kappa$ we have $X\in V[G\restriction \alpha]$. Since $\kappa$ is not collapsed while going from $V$ to $V[G\restriction \alpha]$,
$X\in \mathcal{P}_{\kappa}(\gamma)^ {V[G \restriction\alpha]}$.
By \textbf{Lemma 4.2} (and following the arguments in the proof of \cite[\textbf{Theorem 35}]{Ina2013}), $\hat{X}\in \mathcal{V}'$, where $\mathcal{V}'$ is the fine measure that $\mathcal{U}$ generates on $\mathcal{P}_{\kappa}(\gamma)^{V[G\restriction \alpha]}$. Now $\mathcal{U}\subseteq \mathcal{V}'\subseteq \mathcal{W}$
since $\mathcal{P}_{\kappa}(\gamma)^{V[G\restriction \alpha]}\subseteq \mathcal{P}_{\kappa}(\gamma)^{\mathcal{N}_{1}}$. Consequently $\mathcal{W}$ is fine.
\textbf{Choice function}: Let $\mathcal{N}_{1}\models f:\mathcal{P}_{\kappa}(\gamma)\rightarrow\gamma$ and $\mathcal{N}_{1}\models\forall X \in \mathcal{P}_{\kappa}(\gamma)(f(X)\in X)$. By \textbf{Lemma 4.1}, for some $\alpha<\kappa$ we get $h=f\restriction\mathcal{P}_{\kappa}(\gamma)^V \in V[G \restriction\alpha]$. By \textbf{Lemma 4.3}, we get $Y\in\mathcal{U}$ and $(g:Y\rightarrow\gamma)^V$ such that $V[G\restriction\alpha]\models h\restriction Y = g$. Now by normality of $\mathcal{U}$ in $V$ we get a set $x$ in $\mathcal{U}$ such that $g$ is constant on $x$, and so $h$ is constant on a set in $\mathcal{U}$. Hence, we obtain a set $y$ in $\mathcal{W}$ such that $f$ is constant on $y$.
\end{proof}
\begin{lem}
{\em In $\mathcal{N}_{1}$, $\mathsf{AC_{\kappa}}$ fails.}
\end{lem}
\begin{proof}
Since the cardinality of $\kappa_{\alpha}^{++}$ is preserved in $\mathcal{N}_{1}$ for $\alpha<\kappa$, we can define in $\mathcal{N}_{1}$ the set $X_{\alpha}=\{x\subseteq \kappa_{\alpha}^{++} : x$ codes a well ordering of $(\kappa_{\alpha}^{+++})^{V}$ of order type $\kappa_{\alpha}^{++}\}$. We claim that $\langle X_{\alpha} : \alpha<\kappa\rangle\in \mathcal{N}_{1}$.
The sets $X_{\alpha}$ have fully symmetric names $\dot{X_{\alpha}}$ (any permutation
of a name for an element of $\dot{X_{\alpha}}$ returns a name for an element of $\dot{X_{\alpha}}$). Let $\dot{X}=\{\dot{X_{\alpha}} : \alpha<\kappa\}$. Consequently, $sym (\dot{X})\in \mathcal{F}$, i.e., $\dot{X}$ is symmetric with respect to $\mathcal{F}$. Since all the names appearing in $\dot{X}$ are from $HS$, $\dot{X}\in HS$. Consequently, $\langle X_{\alpha} : \alpha<\kappa\rangle\in \mathcal{N}_{1}$.
Although each $X_{\alpha}\not = \emptyset$, we claim that $(\prod_{\alpha<\kappa}X_{\alpha})^{\mathcal{N}_{1}}=\emptyset$. Otherwise let $y\in(\prod_{\alpha<\kappa}X_{\alpha})^{\mathcal{N}_{1}}$. Since $y$ is a sequence of sets of ordinals, so can be coded as a set of ordinals.
Then there is a $\gamma<\kappa$ such that $y\in V[G\restriction\gamma]$ by \textbf{Lemma 4.1} and $V[G\restriction\gamma]$ is $\mathbb{P}$-generic over $V$ such that $\vert\mathbb{P}\vert<\kappa$. So there is a final segment of the sequence $\langle (\kappa^{+++}_{\alpha}): \alpha<\kappa\rangle$ which remains a sequence of cardinals in $V[G\restriction\gamma]$ which is a contradiction.
\end{proof}
\end{proof}
\begin{remark}
Since $\mathsf{GCH}$ implies $\mathsf{AC}$, $\mathsf{GCH}$ is weakened to a form which states that {\em there is no injection from $\delta^{++}$ into $\mathcal{P}(\delta)$} in \cite[\textbf{Theorem 3}]{Apt2012}. We follow this weakened version of $\mathsf{GCH}$ in our case. We follow the explanation given in \cite[\textbf{section 4.1}]{Kar2019} by Karagila, to see that in $\mathcal{N}_{1}$, $\mathsf{GCH}$ holds for a limit cardinal $\delta$ if and only if $\delta>\kappa$.
The referee suggested us to remark the following. In the context of $\mathsf{ZF}$, there are two reasonable definitions for the statement “$\mathsf{GCH}$ at $\mu$”.
\begin{enumerate}
\item There is no injection $\mu^{++} \rightarrow^{inj} \mathcal{P}(\mu)$.
\item There is no surjection $\mathcal{P}(\mu) \rightarrow^{sur} \mu^{++}$.
\end{enumerate}
In $\mathsf{ZF}$, it is possible that there is no $\mu^{+}\rightarrow^{inj} \mathcal{P}(\mu)$, but there is always a surjection
$\mathcal{P}(\mu) \rightarrow^{sur} \mu^{+}$. In our case the above two definitions behave the
same, so the referee suggested us to remark that both definitions (1) and (2) work, by the same proof.
\end{remark}
\section{Proving Dimitriou's Conjecture}
Fix an arbitrary $n_{0}\in \omega$. Apter \cite[\textbf{Theorem 1}]{Apt1983a} obtained a model of $\mathsf{ZF+\neg DC_{\aleph_{n_{0}+1}}}$ where $\aleph_{\omega}$ carries a Rowbottom filter and $\mathsf{DC_{\aleph_{n_{0}}}}$ holds, from an $\omega$-sequence of measurable cardinals.
In \textbf{section 8}, we observe that there is an alternating sequence of measurable and non-measurable cardinals in that model. Apter constructed the model based on Easton support products of L\'{e}vy collapse.
Consequently, $\mathsf{DC_{\aleph_{n_{0}}}}$ was preserved (cf. \cite[\textbf{Lemma 1.4}]{Apt1983a}). Dimitriou \cite[\textbf{section 1.4}]{Dim2011} constructed a similar model with an alternating sequence of measurable and non-measurable cardinals, excluding the singular limits. She constructed the model based on finite support products of collapsing functions, unlike the model from \cite{Apt1983a}.
In \cite{Dim2011}, Dimitriou claimed that by using such a finite support product construction, a lot of arguments could be made easier. In particular, she used finite support products of {\em injective tree-Prikry forcings}, in several constructions from \cite[\textbf{Chapter 2}]{Dim2011}. There are many models of $\mathsf{ZF}$ constructed using the finite support products of L\'{e}vy collapse. Hayut and Karagila \cite[\textbf{Theorem 5.6}]{HK2019} considered a symmetric extension constructed using the finite support products of L\'{e}vy collapse.
In \textbf{section 6}, we encounter two models of $\mathsf{ZF}$ constructed using the finite support products of L\'{e}vy collapse due to Apter and Cody from \cite{AC2013} (cf. \cite[\textbf{Theorem 2}]{AH1991} as well).
On the other hand, there is a downside to this method.
Specifically, Dimitriou conjectured that $\mathsf{DC_{\omega}}$ would fail in the model.
In this section, we prove that $\mathsf{AC_{\omega}}$ fails in the model and thus prove the conjecture of Dimitriou. In other words, we prove \textbf{Theorem 1.6}. We refer the
reader to the terminologies from \textbf{section 2.4}.
\begin{proof}{\textbf{(of Theorem 1.6)}}
Firstly, we give a description of the symmetric extension constructed in \cite[\textbf{section 1.4}]{Dim2011} as follows.
\begin{enumerate}
\item \textbf{The ground model ($V$).}
Let $V$ be a model of $\mathsf{ZFC}$, $\rho$ be an ordinal, and $\mathcal{K}=\langle\kappa_{\epsilon}:0<\epsilon<\rho\rangle$ be a sequence of measurable cardinals. Let $\kappa_0$ be a regular cardinal below all the measurable cardinals in $\mathcal{K}$.
\item \textbf{Defining the triple $(\mathbb{P},\mathcal{G},\mathcal{I})$.}
\begin{itemize}
\item Let $\kappa_{1}' = \kappa_0$. For each $1<\epsilon< \rho$ we define the following cardinals,
$\kappa_{\epsilon}'= \kappa_{\epsilon-1}^{+}$ if $\epsilon$ is a successor ordinal,
$\kappa_{\epsilon}'= (\bigcup_{\zeta<\epsilon} \kappa_{\zeta})^+ $ if $\epsilon$ is a limit ordinal and $\bigcup_{\zeta<\epsilon} \kappa_{\zeta}$ is singular,
$\kappa_{\epsilon}'= (\bigcup_{\zeta<\epsilon} \kappa_{\zeta})^{++}$ if $\epsilon$ is a limit ordinal and $\bigcup_{\zeta<\epsilon} \kappa_{\zeta}=\kappa_{\epsilon}$ is regular,
$\kappa_{\epsilon}'= \bigcup_{\zeta<\epsilon} \kappa_{\zeta}$ if $\epsilon$ is a limit ordinal and $\bigcup_{\zeta<\epsilon} \kappa_{\zeta}<\kappa_{\epsilon}$ is regular.
Let $\mathbb{P} = \prod_{0<i<\rho} \mathbb{P}_{i}$ be the Easton support product of $\mathbb{P}_{i}=Fn(\kappa_{i}', \kappa_{i},\kappa_{i}')$ ordered componentwise where for each $0<i<\rho$, $Fn(\kappa_{i}', \kappa_{i},\kappa_{i}')$= $\{p:\kappa_{i}'\rightharpoonup\kappa_{i}:\vert p\vert<\kappa_{i}'$ and $p$ is an injection$\}$ ordered by reverse inclusion. We denote by $p:\kappa_{i}'\rightharpoonup\kappa_{i}$ a partial function from $\kappa_{i}'$ to $\kappa_{i}$.
\item Let $\mathcal{G}=
\prod_{0<i<\rho} \mathcal{G}_{i}$ where for each $0<i<\rho$, $\mathcal{G}_{i}$ is the full permutation group of $\kappa_{i}$ that can be extended to $\mathbb{P}_{i}$ by permuting the range of its conditions, i.e., for all $a\in \mathcal{G}_{i}$ and $p\in \mathbb{P}_{i}$, $a(p)=\{(\psi,a(\beta)):(\psi,\beta)\in p\}$.
\item For every finite sequence of ordinals $e=\{\alpha_i: 1\leq i\leq m\}$ such that for every $1\leq i\leq m$ there is a distinct $0<\epsilon_{i}< \rho$ such that $\alpha_{i}\in (\kappa_{\epsilon_i}',\kappa_{\epsilon_i})$, we define $E_{e}=\{\langle \emptyset,... ,p_{\epsilon_1}\cap (\kappa_{\epsilon_1}'\times \alpha_1),\emptyset,...,p_{\epsilon_2}\cap (\kappa_{\epsilon_2}'\times\alpha_2),\emptyset,...,p_{\epsilon_i}\cap (\kappa_{\epsilon_i}'\times\alpha_i),\emptyset,...,p_{\epsilon_m}\cap (\kappa_{\epsilon_m}'\times \alpha_m), \emptyset,...\rangle; \overrightarrow{p}\in \mathbb{P} \}$ and $\mathcal{I}=\{E_e : e\in \prod^{fin}_{0<i<\rho}(\kappa_{i}', \kappa_{i})\}$ where
$\prod^{fin}_{0<i<\rho}(\kappa_{i}', \kappa_{i})$ is the finite support product.
\end{itemize}
\item \textbf{Defining the symmetric extension of $V$.}
Clearly, $\mathcal{I}$ is a projectable symmetry generator with projections $\overrightarrow{p}\restriction E_{e}=\langle \emptyset,..., p_{\epsilon_1}\cap (\kappa_{\epsilon_1}'\times \alpha_1),\emptyset,...,p_{\epsilon_2}\cap (\kappa_{\epsilon_2}'\times\alpha_2),\emptyset,...,p_{\epsilon_m}\cap (\kappa_{\epsilon_m}'\times\alpha_m),\emptyset,...\rangle$. Let $\mathcal{I}$ generate a normal filter $\mathcal{F}_{\mathcal{I}}$ over $\mathcal G$. Let $G$ be a $\mathbb{P}$-generic filter over $V$. We consider the symmetric model $V(G)^{\mathcal{F}_{\mathcal{I}}}$ as our desired symmetric extension.
\end{enumerate}
It is possible to see that $\mathbb{P}$ is $(\mathcal{G},\mathcal{I})$-homogeneous and so $\langle\mathbb{P},\mathcal{G},\mathcal{I}\rangle$ has the approximation property.
Consequently, by \textbf{Lemma 2.16} for all set of ordinals $X\in V(G)^{\mathcal{F}_{\mathcal{I}}}$, there exists an $E_{e}\in \mathcal{I}$ such that $X\in V[G\cap E_{e}]$.
Following \cite[\textbf{Lemma 1.35}]{Dim2011}, for every $0<\epsilon<\rho$, $(\kappa'_{\epsilon})^{+}=\kappa_{\epsilon}$ in $V(G)^{\mathcal{F}_{\mathcal{I}}}$.
We prove that $\mathsf{AC_{\omega}}$ fails in $V(G)^{\mathcal{F}_{\mathcal{I}}}$. For the sake of convenience we call $V(G)^{\mathcal{F}_{\mathcal{I}}}$ as $V(G)$, $HS^{\mathcal{F}_{\mathcal{I}}}$ as $HS$, and $\mathcal{F}_{\mathcal{I}}$ as $\mathcal{F}$.
\begin{lem}
{\em In $V(G)$, $\mathsf{AC_{\omega}}$ fails.}
\end{lem}
\begin{proof}
Since the cardinality of $\kappa_{n}'$ is preserved in $V(G)$ for $n<\omega$, we can define in $V(G)$ the set $X_{n}=\{ x\subseteq \kappa_{n}' : x$ codes a well ordering of $((\kappa_{n}')^{+})^{V}$ of order type $\kappa_{n}'\}$. We claim that $\langle X_{n} : n<\omega\rangle\in V(G)$.
The sets $X_{n}$ have fully symmetric names $\dot{X_{n}}$ (any permutation
of a name for an element of $\dot{X_{n}}$ returns a name for an element of $\dot{X_{n}}$). Let $\dot{X}=\{\dot{X_{n}} : n<\omega\}$. Consequently, $sym (\dot{X})\in \mathcal{F}$, i.e., $\dot{X}$ is symmetric with respect to $\mathcal{F}$. Since all the names appearing in $\dot{X}$ are from $HS$, $\dot{X}\in HS$. Consequently, $\langle X_{n} : n<\omega\rangle\in V(G)$.
Although $X_n\not = \emptyset$, we claim that $(\prod_{n<\omega}X_{n})^{V(G)}=\emptyset$. Otherwise let $y\in(\prod_{n<\omega}X_{n})^{V(G)}$. Since $y$ is a sequence of sets of ordinals, so can be coded as a set of ordinals. Thus, there is an $e=\{\alpha_1,...,\alpha_m\}$ such that $y\in V[G\cap E_{e}]$ by \textbf{Lemma 2.16}. There are distinct $\epsilon_{i}$ such that $\alpha_{i}\in (\kappa_{\epsilon_i}',\kappa_{\epsilon_i})$ and let $l$ be max$\{\epsilon_{i}: \alpha_{i}\in e\}$ such that $l$ is an integer. Next let $M=\{i:\epsilon_{i}\leq l\}$ and $M'=\{i:\epsilon_{i}> l\}$.
Then $V[G\cap E_{e}]$ is $\prod_{i\in M}Fn(\kappa'_{\epsilon_{i}}, \alpha_{i},\kappa'_{\epsilon_{i}})\times \prod_{i\in M'}Fn(\kappa'_{\epsilon_{i}}, \alpha_{i},\kappa'_{\epsilon_{i}})$-generic over $V$. By closure properties of $\prod_{i\in M'}Fn(\kappa'_{\epsilon_{i}}, \alpha_{i},\kappa'_{\epsilon_{i}})$, all elements of the sequence $\langle (\kappa'_{n})^{+}: n<\omega\rangle$ remain cardinals after forcing with $\prod_{i\in M'}Fn(\kappa'_{\epsilon_{i}}, \alpha_{i},\kappa'_{\epsilon_{i}})$.
Next, since $M$ is finite we can find $j<\omega$ such that for all $r\geq j$, $\vert\prod_{i\in M}Fn(\kappa'_{\epsilon_{i}}, \alpha_{i},\kappa'_{\epsilon_{i}})\vert<\kappa_{r}$. Thus, a final segment of the sequence $\langle (\kappa'_{n})^{+}: n<\omega\rangle$ remains a sequence of cardinals in $V[G\cap E_{e}]$ which is a contradiction.
\end{proof}
\end{proof}
\begin{remark}
Hayut and Karagila \cite[\textbf{Theorem 5.6}]{HK2019} proved the following.
\begin{itemize}
\item Assuming the existence of countably many measurable cardinals, it is consistent that there is a uniform ultrafilter on $\aleph_{\omega}$ but for all $0 < n < \omega$, there
are no uniform ultrafilters on $\aleph_{n}$.
\end{itemize}
They considered a symmetric extension $M$ based on finite support product of the symmetric collapses $Col(\kappa_{n}, < \kappa_{n+1})$. Following the proof of \textbf{Lemma 5.1}, we can say that $\mathsf{AC_{\omega}}$ fails in the symmetric extension $M$. We consider another similar symmetric extension. Let $V_{1}$ be a model of $\mathsf{ZFC}$ where $\langle \kappa_{n} : 1\leq n<\omega\rangle$ is a countable sequence of supercompact cardinals. Let $\mathbb{Q}$ be the forcing notion (see \cite{Apt1983, Apt2005}) which makes the supercompactness of each $\kappa_{n}$ indestructible under $\kappa_{n}$-directed closed forcing notions. Let $H$ be a $\mathbb{Q}$-generic filter over $V_{1}$ and $V=V_{1}[H]$ be our ground model. Let $\kappa_{0}=\omega$ in $V$.
Consider the symmetric extension $\mathcal{N}$ obtained by taking the finite support product of
the symmetric collapses $Col(\kappa_{n}, < \kappa_{n+1})$. In the resulting model $\mathcal{N}$ the following hold:
\begin{enumerate}
\item Since the forcing notions involved are weakly homogeneous, if $A$ is a set of ordinals in $\mathcal{N}$, then $A$ was added by an intermediate submodel where $\mathsf{AC}$ holds.
\item For $n > 0$, each $\kappa_{n}$ becomes $\aleph_{n}$ in $\mathcal{N}$.
\end{enumerate}
Following \cite[\textbf{Theorem 4.3}]{HK2019}, we can observe that for each $1\leq n<\omega$, there are no uniform ultrafilters on $\aleph_{n}$ in $\mathcal{N}$. Consequently for each $1\leq n<\omega$, $\aleph_{n}$ can not be a measurable cardinal in $\mathcal{N}$. Since we are considering symmetric extension based on finite support products, $\mathsf{AC_{\omega}}$ fails following the proof of \textbf{Lemma 5.1}.
We can see that each $\aleph_{n}$ remains a Ramsey cardinal for $1 \leq n<\omega$ in $\mathcal{N}$. Fix $1\leq n<\omega$. Let $f:[\kappa_{n}]^{<\omega}\rightarrow 2$ is in $\mathcal{N}$. Since $f$ can be coded by a set of ordinals, $f$ was added by an intermediate submodel (say $V'$) where $\mathsf{AC}$ holds. Without loss of generality, we can say that $V'= V[G_1][G_2]$ where $G_1$ is $\mathbb{Q}_1$-generic over $V$ such that $\mathbb{Q}_1$ is $\kappa_{n}$-directed closed and $G_2$ is $\mathbb{Q}_2$-generic over $V[G_{1}]$ such that $\vert\mathbb{Q}_2\vert<\kappa_{n}$. Since $\mathbb{Q}_{1}$ is $\kappa_{n}$-directed closed, $\kappa_{n}$ remains supercompact in $V[G_{1}]$ as the supercompactness of $\kappa_{n}$ was indestructible under $\kappa_{n}$-directed closed forcing notions in $V$.
Consequently, $\kappa_{n}$ remains a Ramsey cardinal in $V[G_{1}]$. Since $\vert\mathbb{Q}_2\vert<\kappa_{n}$, $\kappa_{n}$ remains Ramsey in $V[G_{1}][G_{2}]$ by \textbf{Theorem 2.5}. There is then a set $X\in [\kappa_{n}]^{\kappa_{n}}$ homogeneous for $f$ in $V'$, and since $V'\subseteq \mathcal{N}$, $X\in[\kappa_{n}]^{\kappa_{n}}$ is homogeneous for $f$ in $\mathcal{N}$. Consequently, for each $1\leq n<\omega$, $\kappa_{n}$ is Ramsey in $\mathcal{N}$.
\end{remark}
\section{Reducing the assumption of supercompactness to strong compactness}
\subsection{Strongly compact Prikry forcing}
Suppose $\lambda>\kappa$ and $\kappa$ be a $\lambda$-strongly compact cardinal in the ground model $V$. Let $\mathcal{U}$ be a $\kappa$-complete fine ultrafilter over $\mathcal{P}_{\kappa}(\lambda)$.
\begin{defn}{(cf. \cite[\textbf{Definition 1.51}]{Git2010})}
A set $T$ is called a {\em $\mathcal{U}$-tree with trunk t} if and only if the following hold.
\begin{enumerate}
\item $T$ consists of finite sequences $\langle P_{1},...,P_{n}\rangle$ of elements of $\mathcal{P}_{\kappa}(\lambda)$ so that $P_{1}\subseteq P_{2}\subseteq ... \subseteq P_{n}$.
\item $\langle T, \unlhd\rangle$ is a tree, where $\unlhd$ is the order of the end extension of finite sequences.
\item $t$ is a trunk of $T$, i.e., $t\in T$ and for every $\eta\in T$, $\eta \unlhd t$ or $t \unlhd \eta$.
\item For every $t\unlhd \eta$, $Suc_{T} (\eta)=\{Q\in \mathcal{P}_{\kappa}(\lambda): \eta \frown\langle Q\rangle\in T\}\in \mathcal{U}$.
\end{enumerate}
\end{defn}
The set $\mathbb{P}_{\mathcal{U}}$ consists of all pairs $\langle t,T\rangle$ such that $T$ is a
$\mathcal{U}$-tree with trunk $t$. If $\langle t,T\rangle, \langle s,S\rangle\in\mathbb{P}_{\mathcal{U}}$, we say that $\langle t,T\rangle$ is stronger than
$\langle s,S\rangle$, and denote this by $\langle t,T\rangle\geq\langle s,S\rangle$, if and only if $T \subseteq S$. We call $\mathbb{P}_{\mathcal{U}}$ with the ordering defined above as the {\em strongly compact Prikry forcing} with respect to $\mathcal{U}$.\footnote{Alternatively, we also recall the definition of a strongly compact Prikry forcing $\mathbb{P}_{\mathcal{U}}$ from \cite{AH1991}. Let $\mathcal{U}$ be a fine measure on $\mathcal{P}_{\kappa}(\lambda)$ and $\mathcal{F}=\{f: f$ is a function from $[\mathcal{P}_{\kappa}(\lambda)]^{<\omega}$ to $\mathcal{U}\}$. In particular, $\mathbb{P}_{\mathcal{U}}$ is the set of all finite sequences of the form $\langle p_{1},...p_{n},f\rangle$ satisfying the following properties.
\begin{itemize}
\item $\langle p_{1},...p_{n}\rangle\in [\mathcal{P}_{\kappa}(\lambda)]^{<\omega}$.
\item for $0\leq i<j\leq n$, $p_{i}\cap \kappa\not= p_{j}\cap \kappa$.
\item $f\in \mathcal{F}$.
\end{itemize}
The ordering on $\mathbb{P}_{\mathcal{U}}$ is given by $\langle q_{1},...q_{m},g\rangle \leq \langle p_{1},...,p_{n},f\rangle$ if and only if we have the following.
\begin{itemize}
\item $n\leq m$.
\item $\langle p_{1},...,p_{n}\rangle$ is the initial segment of $\langle q_{1},...,q_{m}\rangle$.
\item For $i=n+1,..., m$, $q_{i}\in f(\langle p_{1},...,p_{n}, q_{n+1},...,q_{i-1}\rangle)$.
\item For $\overrightarrow{s}\in [\mathcal{P}_{\kappa}(\lambda)]^{<\omega}$, $g(\overrightarrow{s})\subseteq f(\overrightarrow{s})$.
\end{itemize}
For any regular $\delta\in [\kappa,\lambda]$, we denote $r\restriction{\delta}=\{\langle p_{0}\cap \delta,...p_{n}\cap\delta\rangle: \exists f\in \mathcal{F} \left[\langle p_{0},...p_{n},f\rangle\in G\right]\}$. In $V[r\restriction \kappa]\subseteq V[G]$, $\kappa$ is a singular cardinal having cofinality $\omega$. Since any two conditions having the same stems are compatible, i.e. any two conditions of the form $\langle p_{1},...,p_{n},f\rangle$ and $\langle p_{1},...,p_{n},g \rangle$ are compatible., \textbf{$\mathbb{P}_{\mathcal{U}}$ is $(\lambda^{<\kappa})^{+}$-c.c.}
}
Let $G$ be a $\mathbb{P}_{\mathcal{U}}$-generic filter over $V$. We summarize the necessary properties of $\mathbb{P}_{\mathcal{U}}$ from \cite{Git2010}.
\begin{itemize}
\item By a Prikry like lemma and a similar proof as in the ordinary Prikry forcing,\footnote{i.e., the arguments of \cite[\textbf{Lemma 1.9}]{Git2010}.} $\mathbb{P}_{\mathcal{U}}$ does not add new bounded subsets to $\kappa$ (cf. \cite[\textbf{Lemma 1.1}]{AH1991}, \cite[\textbf{Theorem 1.52}]{Git2010}).
\item Every cardinal in $(\kappa, \lambda]$ is collapsed to have size $\kappa$ in $V[G]$ (cf. \cite[\textbf{Lemma 1.50}]{Git2010} and the arguments before \cite[\textbf{Theorem 1.52}]{Git2010}).
\item Every $\delta \in [\kappa, \mu]$ of cofinality $\geq \kappa$ (in $V$) changes its cofinality to $\omega$ in $V[G]$ (cf. \cite[\textbf{Lemma 1.50, Theorem 1.52}]{Git2010}), where $\mu=\lambda$ if $cf(\lambda) \geq \kappa$ and $\mu=\lambda^{+}$ if $cf(\lambda) < \kappa$.
\item Any two conditions with the same trunk, i.e. of the form $\langle t, T\rangle$ and $\langle t, S\rangle$ are compatible. Also there are $\lambda^{<\kappa}$ many possibilities for trunks for members of $\mathbb{P}_{\mathcal{U}}$. Consequently, $\mathbb{P}_{\mathcal{U}}$ satisfies the $(\lambda^{<\kappa})^{+}$-c.c. (cf. \cite[\textbf{Lemma 1.48}]{Git2010} and the arguments before \cite[\textbf{Theorem 1.52}]{Git2010}).
\end{itemize}
\subsection{The Proofs of Observations 1.7 and 1.8} In this subsection, we prove \textbf{Observation 1.7} and \textbf{Observation 1.8}.
\begin{proof}{\textbf{(of Observation 1.7)}} We perform the construction in two stages. In the first stage, we consider a model similar to the choiceless model constructed in \cite[\textbf{section 2}]{AH1991}.
\begin{enumerate}
\item \textbf{The ground model ($V$):} We start with a model $V_{0}$ of $\mathsf{ZFC}$ where $\kappa$ is a strongly compact cardinal, $\theta$ is an ordinal, and $\mathsf{GCH}$ holds. By \cite[\textbf{Theorem 3.1}]{ADU2021} we can obtain a forcing extension $V_{1}$ in which the strong compactness of $\kappa$ is indestructible under $Add(\kappa, \theta)$ for all $\theta$. Then forcing with $Add(\kappa, \theta)$, we may assume without loss of generality that $\kappa$ is
strongly compact and $2^{\kappa}=\theta$ in a forcing extension $V$ of $V_{1}$. Let $\lambda$ be a cardinal in $V$ such that $\kappa < \lambda$ and $(cf(\lambda))^{V}<\kappa$.
\item \textbf{Defining an inner model of a forcing extension of $V$:}
\begin{itemize}
\item Let $\mathcal{U}$ be a fine measure on $\mathcal{P}_{\kappa}(\lambda)$ and $\mathbb{P}=\mathbb{P}_{\mathcal{U}}$ be the strongly compact Prikry forcing. Let $G$ be a $\mathbb{P}_{\mathcal{U}}$-generic filter over $V$.
\item We consider a model similar to the choiceless model constructed in \cite[\textbf{section 2}]{AH1991}. In particular, we consider our model $\mathcal{N}$ to be the least model of $\mathsf{ZF}$ extending $V$ and containing
$r\restriction \delta$ for each regular $\delta\in [\kappa,\lambda)$
where $r\restriction{\delta}=\{\langle p_{0}\cap \delta,...,p_{n}\cap\delta\rangle: \exists f\in \mathcal{F} \left[\langle p_{0},...,p_{n},f\rangle\in G\right]\}$ but not the $\lambda$-sequence of $r\restriction\delta$'s.
\end{itemize}
\end{enumerate}
We follow the homogeneity of strongly compact Prikry forcing mentioned in \cite[\textbf{Lemma 2.1}]{AH1991} to observe the following lemma.
\begin{lem}
{\em If $A\in \mathcal{N}$ is a set of ordinals, then $A\in V[r\restriction \delta]$ for some regular $\delta\in[\kappa,\lambda)$.}
\end{lem}
\begin{lem}
{\em In $\mathcal{N}$, $\kappa$ is a strong limit cardinal.}
\end{lem}
\begin{proof}
Since $V\subseteq \mathcal{N}\subseteq V[G]$ and $\mathbb{P}$ does not add bounded subsets to $\kappa$, $V$ and $\mathcal{N}$ have the same bounded subsets of $\kappa$.\footnote{We can observe another argument from \cite[\textbf{Lemma 2.2}]{AH1991}.} Consequently, in $\mathcal{N}$, $\kappa$ is a limit of inaccessible cardinals and thus a strong limit cardinal as well.
\end{proof}
As explained in the introduction, our definitions of “strong limit cardinal” and “inaccessible cardinal” generally do not make sense in choiceless models. In spite of that, we can see that the assertion in \textbf{Lemma 6.3} makes sense (see the paragraph after \cite[\textbf{Theorem 1}]{AC2013}). Since $\mathcal{N}$ and $V$ have the same bounded subsets of $\kappa$, the usual definitions of “$\kappa$ is a strong limit cardinal”
and “$\delta < \kappa$ is an inaccessible cardinal” make sense in $\mathcal{N}$.
\begin{lem}
{\em If $\gamma\geq\lambda$ is a cardinal in $V$, then $\gamma$ remains a cardinal in $\mathcal{N}$.}
\end{lem}
\begin{proof}
For the sake of contradiction, let $\gamma$ is not a cardinal in $\mathcal{N}$. There is then a bijection $f:\alpha\rightarrow\gamma$ for some $\alpha<\gamma$ in $\mathcal{N}$. Since $f$ can be coded by a set of ordinals, by \textbf{Lemma 6.2} we have $f\in V[r\restriction\delta]$ for some regular $\delta\in [\kappa,\lambda)$. Since $\mathsf{GCH}$ is assumed in $V_{0}$ we have $(\delta^{<\kappa})^{V_{0}}=\delta$, and since $Add(\kappa,\theta)$ preserves cardinals and adds no sequences of ordinals of length less than $\kappa$, we conclude that $(\delta^{<\kappa})^{V}=(\delta^{<\kappa})^{V_{0}}=\delta$.
Now $\mathbb{P}_{\mathcal{U}\restriction \delta}$ is $(\delta^{<\kappa})^{+}$-c.c. in $V$ and hence $\delta^{+}$-c.c. in $V$. Consequently, $\gamma$ is a cardinal in $V[r\restriction\delta]$ which is a contradiction.
\end{proof}
\begin{lem}
{\em In $\mathcal{N}$, $cf(\kappa)=\omega$. Moreover, $(\kappa^{+})^{\mathcal{N}}=\lambda$ and $cf(\lambda)^{\mathcal{N}} = cf(\lambda)^V$.}
\end{lem}
\begin{proof}
For each regular $\delta\in [\kappa,\lambda)$, we have $V[r\restriction \delta]\subseteq \mathcal{N}$. Consequently, $cf(\kappa)^{\mathcal{N}}=\omega$ since $cf(\kappa)^{V[r\restriction \kappa]}=\omega$.
Following \cite[\textbf{Lemma 2.4}]{AH1991}, every ordinal in $(\kappa,\lambda)$ which is a cardinal in $V$ collapses to have size $\kappa$ in $\mathcal{N}$, and so $(\kappa^{+})^{\mathcal{N}}=\lambda$.
Since $V$ and $\mathcal{N}$ have same bounded subsets of $\kappa$, we see that $cf(\lambda)^{\mathcal{N}}=cf(\lambda)^{V}<\kappa$.
\end{proof}
We can see that since, $V\subseteq \mathcal{N}$ and $(2^{\kappa}=\theta)^{V}$, there is a $\theta$-sequence of distinct subsets of $\kappa$ in $\mathcal{N}$. Since $cf(\kappa^{+})^{\mathcal{N}}<\kappa$ we can also see that $\mathsf{AC_{\kappa}}$ fails in $\mathcal{N}$.
In the second stage, we consider an inner model of a forcing extension of $\mathcal{N}$ based on a product of L\'{e}vy collapses as done in the proof of \cite[\textbf{Theorem 2}]{AC2013}.
\begin{enumerate}
\item \textbf{The ground model ($\mathcal{N}$).}
Consider the ground model to be $\mathcal{N}$. Let $\langle \kappa_{n}:n<\omega\rangle$ be a sequence of inaccessible cardinals less than $\kappa$ which is cofinal in $\kappa$.
\item \textbf{Defining an inner model of a forcing extension of $\mathcal{N}$.}
\begin{itemize}
\item Let $\mathbb{P}=Col(\omega,<\kappa)$, and let $G$ be $\mathbb{P}$-generic over $\mathcal{N}$. Let $\mathbb{P}_{n}=Col(\omega,<\kappa_{n})$. Following the proof of \cite[\textbf{Theorem 2}]{AC2013}, $G_{n}=G\cap \mathbb{P}_{n}$ is $\mathbb{P}_{n}$-generic over $\mathcal{N}$.
\item Let $\mathcal{M}$ be the least model of $\mathsf{ZF}$ extending $\mathcal{N}$ containing each $G_{n}$, but not $G$ as constructed in \cite[\textbf{Theorem 2}]{AC2013}.
\end{itemize}
\end{enumerate}
Following the proof of \cite[\textbf{Theorem 2}]{AC2013}, we have the following in $\mathcal{M}$.
\begin{enumerate}
\item Since $\mathcal{M}$ contains $G_{n}$ for each $n$, cardinals in $[\omega,\kappa)$ are collapsed to have size $\omega$ and so $\aleph_{1}^{\mathcal{M}}\geq\kappa$.
\item If $x \in \mathcal{M}$ is a set of
ordinals, then $x \in \mathcal{N}[G_{n}]$ for some $n < \omega$.
\item Since $Col(\omega, <\kappa_{n})$ is canonically well-orderable in $\mathcal{N}$ with order type $\kappa_{n}$, cardinals and cofinalities greater than or equal to $\kappa$ are preserved to $\mathcal{N}[G_{n}]$.
\item Since $\kappa$ is not collapsed, $\kappa=\aleph_{1}^{\mathcal{M}}$, $cf(\aleph_{1})^{\mathcal{M}}=cf(\aleph_{2})^{\mathcal{M}}=\omega$. Consequently, $\mathsf{AC_{\omega}}$ fails in $\mathcal{M}$.
\item There is a sequence of distinct subsets of $\aleph_{1}$ of length $\theta$.
\end{enumerate}
\end{proof}
\begin{proof}{\textbf{(of Observation 1.8)}} We recall the model $\mathcal{N}$ from the previous proof. We consider an inner model of the forcing extension of $\mathcal{N}$ as done in the proof of \cite[\textbf{Theorem 3}]{AC2013}.
\begin{enumerate}
\item \textbf{The ground model ($\mathcal{N}$).}
Consider the ground model to be $\mathcal{N}$ as in the previous proof. Let $\langle \kappa_{n}:n<\omega\rangle$ be a sequence of inaccessible cardinals less than $\kappa$ which is cofinal in $\kappa$.
\item \textbf{Defining an inner model of a forcing extension of $\mathcal{N}$.}
\begin{itemize}
\item Let $\mathbb{P}_{0}=Col(\omega,<\kappa_{0})$, $\mathbb{P}_{i}=Col(\kappa_{i-1},<\kappa_{i})$ for $1\leq i<\omega$. Let $\mathbb{P}=\Pi_{i<\omega}^{fin} \mathbb{P}_{i}$ be a finite support product of $\mathbb{P}_{i}$. For each $n<\omega$, we can factor $\mathbb{P}$ as $\mathbb{P}\cong \mathbb{P}_{n}^{*}\times \mathbb{P}^{n}$ where $\mathbb{P}_{n}^{*}=\Pi_{0\leq i\leq n}^{fin} \mathbb{P}_{i}$ and $\mathbb{P}^{n}=\Pi_{n+1\leq i<\omega}^{fin} \mathbb{P}_{i}$. Let $G\cong G_{n}^{*}\times G^{n}$ be a $\mathbb{P}$-generic filter over $\mathcal{N}$.
Following \cite[\textbf{Theorem 3}]{AC2013}, each $G_{n}^{*}$ is $\mathbb{P}_{n}^{*}$-generic over $\mathcal{N}$.
\item Let $\mathcal{M}$ be the least model of $\mathsf{ZF}$ extending $\mathcal{N}$ containing each $G_{n}^{*}$, but not $\langle G^{*}_{n}:n<\omega \rangle$ as constructed in \cite[\textbf{Theorem 3}]{AC2013}.
\end{itemize}
\end{enumerate}
Following the proof of \cite[\textbf{Theorem 3}]{AC2013}, we have the following in $\mathcal{M}$.
\begin{enumerate}
\item Since $G^{*}_{n} \in \mathcal{M}$ for each $n < \omega$, we have $\aleph_{\omega} \geq \kappa$ and hence $\aleph_{\omega+1} \geq (\kappa^{+})^{\mathcal{N}}$ in $\mathcal{M}$.
\item If $x$ is a set of ordinals in $\mathcal{M}$, then $x \in \mathcal{N}[G^{*}_{n}]$ for some
$n < \omega$ (cf. \cite[\textbf{Lemma 6}]{AC2013}).
\item Since $\mathcal{N}$
and $V$ contain the same bounded subsets of $\kappa$, and $V \subseteq \mathcal{N}$, $\mathbb{P}^{*}_{n}$
can be well-ordered in both $V$ and $\mathcal{N}$ with order type less than $\kappa$. Therefore,
cardinals and cofinalities greater
than or equal to $\kappa$ are preserved.
\item $\kappa=\aleph_{\omega}$ and $(\kappa^{+})^{\mathcal{N}}=\aleph_{\omega+1}$ are both singular with $\omega\leq cf(\aleph_{\omega+1})<\aleph_{\omega}$.
\item $\mathsf{AC_{\omega}}$ fails in $\mathcal{M}$.
\item There is a sequence of distinct subsets of $\aleph_{\omega}$ of length $\theta$.
\end{enumerate}
\end{proof}
\section{Infinitary Chang conjecture from a measurable cardinal}
\subsection{Infinitary Chang conjecture}
We recall the required definitions and relevant lemmas from \cite[\textbf{Chapter 3}]{Dim2011}. For the sake of our convenience we denote a structure $\mathcal{A}$ on domain $A$ as $\mathcal{A}=\langle A,...\rangle$.
\begin{defn}
{(\textbf{Set of good indiscernibles}; \cite[\textbf{Definition 3.2}]{Dim2011})} For a structure $\mathcal{A}=\langle A,...\rangle$ with $A\subseteq Ord$, a set $I\subseteq A$ is a {\em set of indiscernibles} if for all $n<\omega$, all $n$-ary formula $\phi$ in the language for $\mathcal{A}$, and every $\alpha_1,...,\alpha_n,\alpha'_1,...,\alpha'_n$ in $I$, if $\alpha_1<...<\alpha_n$ and $\alpha'_1<...<\alpha'_n$ then
\centerline{$\mathcal{A}\models \phi(\alpha_1,...,\alpha_n)$ if and only if $\mathcal{A}\models \phi(\alpha'_1,...,\alpha'_n)$.}
The set $I$ is a {\em set of good indiscernibles} if and only if it is a set of indiscernibles and we allow parameters that lie below min$\{\alpha_1,...,\alpha_n,\alpha'_1,...,\alpha'_n \}$, i.e., if for every $x_1,...,x_m\in A$ such that $x_1,...,x_m\leq min\{\alpha_1,...,\alpha_n,\alpha'_1,...,\alpha'_n\}$ and every $(n+m)$-ary formula $\phi$,
\centerline{$\mathcal{A}\models \phi(x_1,...,x_m,\alpha_1,...,\alpha_n)$ if and only if $\mathcal{A}\models \phi(x_1,...,x_m,\alpha'_1,...,\alpha'_n)$.}
\end{defn}
\begin{defn}
{(\textbf{$\alpha$-Erd\H{o}s cardinal}; \cite[\textbf{Definition 0.14}]{Dim2011})} The partition relation $\alpha\rightarrow (\beta)^{\gamma}_{\delta}$ for ordinals $\alpha,\beta,\gamma,\delta$ means for all $f:[\alpha]^{\gamma}\rightarrow\delta$ there is an $X\in[\alpha]^{\beta}$ such that $X$ is homogeneous for $f$. For an infinite ordinal $\alpha$, the {\em $\alpha$-Erd\H{o}s cardinal} $\kappa(\alpha)$ is the least $\kappa$ such that $\kappa\rightarrow(\alpha)^{<\omega}_{2}$.
\end{defn}
For cardinals $\kappa>\lambda$ and ordinal $\theta<\kappa$ we mean $\kappa\rightarrow^{\theta}(\lambda)^{<\omega}_{2}$ if for every first order structure $\mathcal{A}=\langle \kappa,...\rangle$ with a countable language, there is a set $I\in [\kappa\backslash\theta]^{\lambda}$ of good indiscernibles for $\mathcal{A}$ (cf. \cite[\textbf{Definition 3.7}]{Dim2011}).
\begin{defn}
{(\textbf{Infinitary Chang conjecture}; \cite[\textbf{Definition 3.10}]{Dim2011})}
Let $\langle \kappa_{n}: n < \omega\rangle$ and $\langle \lambda_{n}: n < \omega\rangle$ be two increasing sequences of cardinals such that $\kappa_{n}>\lambda_{n}$ for every $n<\omega$. The {\em infinitary Chang conjecture} for these sequences, written as $(\kappa_n)_{n\in\omega}\twoheadrightarrow (\lambda_n)_{n\in\omega}$, is the statement ``for every first order structure $\mathcal{A}=\langle \bigcup_{n<\omega}\kappa_{n},... \rangle$ there is an elementary substructure $\mathcal{B}\prec\mathcal{A}$ with domain $B$ of cardinality $\bigcup_{n<\omega}{\lambda_{n}}$ such that for every $n\in\omega$, $\vert B\cap \kappa_n\vert=\lambda_{n}$".
\end{defn}
\begin{defn}{(cf. \cite[\textbf{Definition 3.14}]{Dim2011})}
Let $\langle \kappa_{i}: i<\omega\rangle$ and $\langle \lambda_{i}: 0< i<\omega\rangle$ be two increasing sequences of cardinals. Let $\kappa=\bigcup_{i<\omega} \kappa_{i}$. We say $\langle\kappa_{i}: i<\omega\rangle$ is a {\em coherent sequence of cardinals} with the property $\kappa_{i+1}\rightarrow^{\kappa_i}(\lambda_{i+1})^{<\omega}_{2}$ if and only if for every structure $\mathcal{A}=\langle \kappa,...\rangle$ with a countable language, there is a $\langle \lambda_{i}: 0< i<\omega\rangle$-coherent sequence of good indiscernibles for $\mathcal{A}$ with respect to $\langle \kappa_{i}: i<\omega\rangle$, i.e., a sequence $\langle A_{i} : 0 < i < \omega\rangle$ with the following properties.
\begin{enumerate}
\item for every $0<i<\omega$, $A_{i} \in
[\kappa_{i}\backslash \kappa_{i-1}]^{\lambda_{i}}$,
\item if $x, y \in [\kappa]^{<\omega}$ are such that $x = \{x_{1},..., x_{n}\}$, $y = \{y_{1},..., y_{n}\}$, $x, y \subseteq \bigcup_{0<i<\omega} A_{i}$, and for every $0<i<\omega$, $\vert x \cap A_{i}\vert = \vert y \cap A_{i}\vert$ then for every $(n + l)$-ary formula $\phi$ in the language of $\mathcal{A}$ and every $z_{1},...,z_{l} < min \{x_{1},...,x_{n},y_{1},...,y_{n}\}$,
$\mathcal{A} \models \phi(z_{1},...,z_{l},x_{1},...,x_{n}) \iff \mathcal{A} \models \phi(z_{1},...,z_{l},y_{1},..., y_{n})$.
\end{enumerate}
\end{defn}
\begin{lem}
{($\mathsf{ZF}$; \textbf{Dimitriou}; \cite[\textbf{Corollary 3.15}]{Dim2011})}{\em Let $\langle \kappa_{i}: i<\omega\rangle$ and $\langle \lambda_{i}: 0< i<\omega\rangle$ be two increasing sequences of cardinals. Let $\kappa=\bigcup_{i<\omega} \kappa_{i}$. If $\langle\kappa_{i}: i<\omega\rangle$ is a coherent sequence of cardinals with the property $\kappa_{i+1}\rightarrow^{\kappa_i}(\lambda_{i+1})^{<\omega}_{2}$ then the infinitary Chang Conjecture $(\kappa_n)_{n\in\omega}\twoheadrightarrow (\lambda_n)_{n\in\omega}$ holds.}
\end{lem}
\begin{lem}
{(\textbf{Dimitriou}; \cite[\textbf{Proposition 3.50}]{Dim2011})} {\em Let us assume that $V\models$ $\mathsf{ZFC}+``\kappa=\kappa(\lambda)$ exists", $\mathbb{P}$ is a forcing notion such that $\vert \mathbb{P}\vert<\kappa$, and $\mathbb{Q}$ is a forcing notion that doesn't add subsets to $\kappa$. If $G$ is $\mathbb{P}\times\mathbb{Q}$ generic then for every $\theta<\kappa$, $V[G]\models\kappa\rightarrow^{\theta}(\lambda)^{<\omega}_{2}$.}
\end{lem}
\begin{lem}
{(\textbf{Dimitriou}; \cite[\textbf{Lemma 3.52}]{Dim2011})} {\em Let $\langle \kappa_{i}: i<\omega\rangle$ and $\langle \lambda_{i}: 0< i<\omega\rangle$ be two increasing sequences of cardinals such that $\langle \kappa_{i}: 0< i<\omega\rangle$ is a coherent sequence of Erd\H{o}s cardinals with respect to $\langle \lambda_{i}: 0< i<\omega\rangle$. If $\mathbb{P}_1$ is a forcing notion of cardinality $<\kappa_1$ and G is $\mathbb{P}_1$-generic, then in $V[G]$, $\langle \kappa_{i}: i<\omega\rangle$ is a coherent sequence of cardinals with the property $\kappa_{i+1}\rightarrow^{\kappa_i}(\lambda_{i+1})^{<\omega}_{2}$.}
\end{lem}
\subsection{The Proofs of Theorems 1.9 and 1.10} In this subsection, we prove \textbf{Theorem 1.9} and \textbf{Theorem 1.10}.
\begin{proof}{\textbf{(of Theorem 1.9)}}
\begin{enumerate}
\item \textbf{The ground model ($V$).}
Let $\kappa$ be a measurable cardinal in a model $V'$ of $\mathsf{ZFC}$. By Prikry forcing it is possible to make $\kappa$ singular with cofinality $\omega$ where an end segment $\langle \kappa_{i}: 1\leq i<\omega\rangle$ of the Prikry sequence $\langle \delta_{i}: 1\leq i <\omega\rangle$ is a {\em coherent} sequence of Ramsey cardinals by \cite[\textbf{Theorem 3}]{AK2006}. Now Ramsey cardinals $\kappa_{i}$ are exactly the $\kappa_{i}$-Erd\H{o}s cardinals. Thus we obtain a generic extension (say $V$) where $\langle\kappa_i: 1\leq i<\omega\rangle$ is a coherent sequence of cardinals with supremum $\kappa$ such that for all $1\leq i<\omega$, $\kappa_i=\kappa(\kappa_i)$. We define the following cardinals.
\begin{enumerate}
\item $\kappa'_{0}=\omega$ and $\kappa_{0}=\aleph_{\omega}$.
\item $\kappa'_{1}=\aleph_{\omega+1}$.
\item $\kappa'_{i}=\kappa_{i-1}^{+\omega_{i-1}+1}$for each $1< i<\omega$.
\end{enumerate}
\item \textbf{Defining the triple $\langle\mathbb{P},\mathcal{G},\mathcal{I}\rangle$.}
We consider a triple similar to the one constructed in \textbf{section 5}.
\begin{itemize}
\item Let
$\mathbb{P} =\prod_{i<\omega}\mathbb{P}_{i}$ be the Easton support product\footnote{In this case, it is equivalent to full
support.} of $\mathbb{P}_{i}=Fn(\kappa_{i}', \kappa_{i},\kappa_{i}')$ ordered componentwise where for each $i<\omega$, $Fn(\kappa_{i}', \kappa_{i},\kappa_{i}')$= $\{p:\kappa_{i}'\rightharpoonup\kappa_{i}:\vert p\vert<\kappa_{i}'$ and $p$ is an injection$\}$ ordered by reverse inclusion.
\item
$\mathcal{G}=
\prod_{i<\omega} \mathcal{G}_{i}$ where for each $i<\omega$, $\mathcal{G}_{i}$ is the full permutation group of $\kappa_{i}$ that can be extended to $\mathbb{P}_{i}$ by permuting the range of its conditions, i.e., for all $a\in \mathcal{G}_{i}$ and $p\in \mathbb{P}_{i}$, $a(p)=\{(\psi,a(\beta)):(\psi,\beta)\in p\}$.
\item
For $m\in\omega$ and $e=\{\alpha_1,...,\alpha_m\}$ a sequence of ordinals such that for each $1\leq i\leq m$, there is a distinct $\epsilon_{i}<\omega$ such that $\alpha_{i}\in (\kappa'_{\epsilon_i},\kappa_{\epsilon_i})$,
we define $E_{e}=\{\langle \emptyset,...,p_{\epsilon_1}\cap (\kappa'_{\epsilon_1}\times \alpha_1),\emptyset,...,p_{\epsilon_2}\cap (\kappa'_{\epsilon_2}\times\alpha_2),\emptyset,...,p_{\epsilon_m}\cap (\kappa'_{\epsilon_m}\times\alpha_m),\emptyset,...\rangle; \overrightarrow{p}\in \mathbb{P} \}$ and $\mathcal{I}=\{E_e : e\in \prod^{fin}_{i<\omega}(\kappa'_{i}, \kappa_{i})\}$.
\end{itemize}
\item \textbf{Defining symmetric extension of $V$.}
Let $\mathcal{I}$ generate a normal filter $\mathcal{F}_{\mathcal{I}}$ over $\mathcal G$.
Let $G$ be a $\mathbb{P}$-generic filter. We consider the symmetric model $V(G)^{\mathcal{F}_I}$. We denote $V(G)^{\mathcal{F}_I}$ by $V(G)$ for the sake of convenience.
\end{enumerate}
Since the forcing notions involved are weakly homogeneous, the following holds.
\begin{lem}
{\em If $A\in V(G)$ is a set of ordinals, then $A\in V[G\cap E_{e}]$ for some $E_{e}\in\mathcal{I}$.}
\end{lem}
Following the arguments in \cite[\textbf{Lemma 1.35}]{Dim2011}, we can see that in $V(G)$, $(\kappa'_{i})^{+}=\kappa_{i}$ for every $i<\omega$.
Similar to the arguments from the proof of \cite[\textbf{Theorem 11}]{AK2006}, it is possible to see that in $V(G)$, $\kappa=\aleph_{(\aleph_{\omega})^{V}}$ and $(\aleph_{\omega})^{V}=\aleph_{1}$. Consequently $\kappa=\aleph_{\omega_{1}}$ and $cf(\kappa)=\omega$ in $V(G)$. Further $\omega_{1}$ is singular in $V(G)$. Following \textbf{Fact 2.19}, $\mathsf{AC_{\omega}}$ fails in $V(G)$. We prove that an infinitary Chang conjecture holds in $V(G)$.
\begin{lem}
{\em In $V(G)$, an infinitary Chang conjecture holds.}
\end{lem}
\begin{proof}
Let $\mathcal{A}=\langle \kappa,...\rangle$ be a structure in a countable language in $V(G)$.
Let $\{\phi_{n}: n<\omega\}$ be an enumeration of the formulas of the language of ${\mathcal{A}}$ such that each $\phi_{n}$ has $k(n)\leq n$ many free variables. Define $f:[\kappa]^{<\omega}\rightarrow 2$ by,
\centerline{$f(\epsilon_{1},...,\epsilon_{n})=1$ if and only if $\mathcal{A}\models\phi_{n}(\epsilon_{1},...,\epsilon_{k(n)})$ and $f(\epsilon_{1},...,\epsilon_{n})=0$ otherwise.}
By \textbf{Lemma 7.8}, there is an $E_{e}\in \mathcal{I}$ such that $f\in V[G\cap E_{e}]$. Fix an arbitrary $1\leq i< \omega$.
We can write $V[G\cap E_{e}]$=$V[G_1][G_2]$ where $G_1$ is $\mathbb{Q}_1$-generic over $V$ such that $\vert\mathbb{Q}_1\vert<\kappa_{i}$, and $G_2$ is $\mathbb{Q}_2$-generic over $V[G_1]$ such that $G_2$ adds no subsets of $\kappa_{i}$. Consequently, by \textbf{Lemma 7.6}, $\kappa_{i}\rightarrow^{\kappa_{i-1}}(\kappa_{i})^{<\omega}_{2}$ in $V[G\cap E_{e}]$.
So, for all $1\leq i<\omega$, $\kappa_{i}\rightarrow^{\kappa_{i-1}}(\kappa_{i})^{<\omega}_{2}$ in $V[G\cap E_{e}]$.
Let $e=\{\alpha_{1},...,\alpha_{m}\}$ where for each $i\in \{1,...,m\}$, there is a dictinct $\epsilon_{i}$ such that $\alpha_{i}\in (\kappa'_{\epsilon_{i}},\kappa_{\epsilon_{i}})$. Consider $j$ to be $max\{\epsilon_{i}:\alpha_{i}\in e\}$. If $G\cap E_{e}$ is $\mathbb{P}$-generic over $V$ then since $\vert \mathbb{P} \vert<\kappa_{j}$,
by \textbf{Lemma 7.7}, $\langle \kappa_{i}:j-1\leq i<\omega\rangle$ is a coherent sequence of cardinals with the property $\kappa_{i}\rightarrow^{\kappa_{i-1}}(\kappa_{i})^{<\omega}_{2}$ for all $j\leq i<\omega$.
By \textbf{Definition 7.4}, there is a $\langle\kappa_{i}:j\leq i<\omega\rangle$-coherent sequence $\langle A_{n}:j\leq n<\omega\rangle$ of good indiscernibles for $\mathcal{A}$ with respect to $\langle\kappa_{i}:j-1\leq i<\omega\rangle$.
We obtain a $\langle\kappa_{i}:j-1\leq i<\omega\rangle$-coherent sequence $\langle A_{n}:j-1\leq n<\omega\rangle$ of good indiscernibles for $\mathcal{A}$ with respect to $\langle\kappa_{i}:j-2\leq i<\omega\rangle$ as follows.
\begin{itemize}
\item Since $\kappa_{j-1}\rightarrow^{\kappa_{j-2}}(\kappa_{j-1})^{<\omega}_{2}$, we obtain a set $A_{j-1}\in [\kappa_{j-1}\backslash \kappa_{j-2}]^{\kappa_{j-1}}$ of indiscernibles for $\mathcal{A}$ with respect to parameters below $\kappa_{j-2}$. Consequently, we obtain a $\langle\kappa_{i}:j-1\leq i<\omega\rangle$-coherent sequence $\langle A_{n}:j-1\leq n<\omega\rangle$ of good indiscernibles for $\mathcal{A}$ with respect to $\langle\kappa_{i}:j-2\leq i<\omega\rangle$.
\end{itemize}
If we continue in this manner step by step for the remaining cardinals $\kappa_{1},...\kappa_{j-2}$, then since $\kappa_{i}\rightarrow^{\kappa_{i-1}}(\kappa_{i})^{<\omega}_{2}$ for each $1\leq i\leq j-2$, we can obtain a $\langle\kappa_{i}:0 < i<\omega\rangle$-coherent sequence $A=\langle A_{n}:0< n<\omega\rangle$ of good indiscernibles for $\mathcal{A}$ with respect to $\langle\kappa_{i}: i<\omega\rangle$ and $A\in V[G\cap E_{e}]\subseteq V(G)$.
Therefore for all $1\leq i<\omega$, $\kappa_{i}\rightarrow^{\kappa_{i-1}}(\kappa_{i})^{<\omega}_{2}$ and $\langle\kappa_{i}:1\leq i<\omega\rangle$ is a coherent sequence of cardinals in $V(G)$ by \textbf{Definition 7.4}. Using \textbf{Lemma 7.5}, we can obtain an infinitary Chang conjecture in $V(G)$ as \textbf{Lemma 7.5} can be proved in $\mathsf{ZF}$.
\end{proof}
\end{proof}
\begin{proof}{\textbf{(of Theorem 1.10)}}
Let $\mathcal{N}$ be the choiceless model constructed in \cite[\textbf{Theorem 11}]{AK2006}.
We first translate the arguments in terms of a symmetric extension based on a symmetric system $\langle \mathbb{P}, \mathcal{G}, \mathcal{F}\rangle$.
\begin{itemize}
\item Consider $V$, $\mathbb{P}$, and $\mathcal{G}$ as mentioned in the previous construction (used for proving \textbf{Theorem 1.9}).
\item Let $\mathcal{I}=\{E_e : e\in \prod_{i< \omega}(\kappa_{i}', \kappa_{i})\}$ where for every $e=\{\alpha_i: i< \omega\}\in \prod_{i\in \omega}(\kappa_{i}', \kappa_{i})$,
$E_{e}=\{\langle p_{i}\cap (\kappa_{i}'\times \alpha_{i}): i<\omega\rangle: \overrightarrow{p}\in \mathbb{P}\}$.
Let $\mathcal{I}$ generate a normal filter $\mathcal{F}_{\mathcal{I}}$ over $\mathcal G$. We define $\mathcal{F}$ to be $\mathcal{F}_{\mathcal{I}}$.
\end{itemize}
Let $G$ be a $\mathbb{P}$-generic filter. We consider the symmetric model $V(G)^{\mathcal{F}}$. We denote $V(G)^{\mathcal{F}}$ by $V(G)$ for the sake of convenience. The model $V(G)$ is analogous to the choiceless model $\mathcal{N}$ constructed in \cite[\textbf{Theorem 11}]{AK2006}.
Since the forcing notions involved are weakly homogeneous, the following holds.
\begin{lem}
{\em If $A\in V(G)$ is a set of ordinals, then $A\in V[G\cap E_{e}]$ for some $E_{e}\in\mathcal{I}$.}
\end{lem}
Similar to \textbf{Lemma 7.9}, we observe an infinitary Chang conjecture in $V(G)$.
\begin{lem}
{\em In $V(G)$, an infinitary Chang conjecture holds .}
\end{lem}
\begin{proof}
Let $\mathcal{A}=\langle \kappa,...\rangle$ be a structure in a countable language in $V(G)$.
Let $\{\phi_{n}: n<\omega\}$ be an enumeration of the formulas of the language of $\mathcal{A}$ such that each $\phi_{n}$ has $k(n)\leq n$ many free variables. Define $f:[\kappa]^{<\omega}\rightarrow 2$ by,
\centerline{$f(\epsilon_{1},...,\epsilon_{n})=1$ if and only if $\mathcal{A}\models\phi_{n}(\epsilon_{1},...,\epsilon_{k(n)})$ and $f(\epsilon_{1},...,\epsilon_{n})=0$ otherwise.}
By \textbf{Lemma 7.10}, there is an $E_{e}\in \mathcal{I}$ such that $f\in V[G\cap E_{e}]$. Fix an arbitrary $1\leq i< \omega$.
We can write $V[G\cap E_{e}]$=$V[G_1][G_2]$ where $G_1$ is $\mathbb{Q}_1$-generic over $V$ such that $\vert\mathbb{Q}_1\vert<\kappa_{i}$, and $G_2$ is $\mathbb{Q}_2$-generic over $V[G_1]$ such that $G_2$ adds no subsets of $\kappa_{i}$. Consequently, by \textbf{Lemma 7.6}, $\kappa_{i}\rightarrow^{\kappa_{i-1}}(\kappa_{i})^{<\omega}_{2}$ in $V[G\cap E_{e}]$.
So, for all $1\leq i<\omega$, $\kappa_{i}\rightarrow^{\kappa_{i-1}}(\kappa_{i})^{<\omega}_{2}$ in $V[G\cap E_{e}]$.
Thus, we obtain a set $A_{i}\in [\kappa_{i}\backslash \kappa_{i-1}]^{\kappa_{i}}$ of good indiscernibles for $\mathcal{A}$ for each $1\leq i<\omega$, in $V[G\cap E_{e}]$. Consequently, we obtain a $\langle\kappa_{i}:0 < i<\omega\rangle$-coherent sequence $A=\langle A_{i}:0<i<\omega\rangle$ of good indiscernibles for $\mathcal{A}$ with respect to $\langle\kappa_{i}: i<\omega\rangle$ and $A\in V[G\cap E_{e}]\subseteq V(G)$. The rest is the same as in the proof of \textbf{Lemma 7.9}.
\end{proof}
Applying \cite[\textbf{Theorem 4}]{AK2006} and \cite[\textbf{Proposition 1}]{AK2008}, we prove that $\aleph_{\omega_{1}}$ is an almost Ramsey cardinal in $V(G)$.
\begin{lem}
{\em In $V(G)$, $\aleph_{\omega_{1}}$ is an almost Ramsey cardinal.}
\end{lem}
\begin{proof}
Following the terminologies from the proof of \cite[\textbf{Theorem 11}]{AK2006} we have $\kappa=\aleph_{\omega_{1}}$ in $V(G)$. We show that $\kappa$ is an almost Ramsey cardinal in $V(G)$.
Let $f:[\kappa]^{<\omega}\rightarrow 2$ be in $V(G)$. Since $f$ can be coded by a set of ordinals, $f\in V[G\cap E_{e}]$ for some $E_{e}\in\mathcal{I}$ by \textbf{Lemma 7.10}. Now, in $V$, $\kappa$ is the supremum of a coherent sequence of Ramsey cardinals $\langle\kappa_{i}: i<\omega\rangle$.
By \cite[\textbf{Theorem 4}]{AK2006} we can see that $\langle\kappa_{i}: i<\omega\rangle$ stays a coherent sequence of Ramsey cardinals in $V[G\cap E_{e}]$.
Also $\kappa$ is the supremum of $\langle\kappa_{i}:i<\omega\rangle$ in $V[G\cap E_{e}]$.
Thus $\kappa$ is an almost Ramsey cardinal in $V[G\cap E_{e}]$ by \cite[\textbf{Proposition 1}]{AK2008}.
Thus for all $\beta<\kappa$, there is a set $X_{\beta}\in V[G\cap E_{e}]\subseteq V(G)$ which is homogeneous for $f$ and has order type at least $\beta$. Hence, $\kappa$ is almost Ramsey in $V(G)$ since $f$ was arbitrary.
\end{proof}
\end{proof}
\section{Mutual stationarity property from a sequence of measurable cardinals}
\subsection{Mutual stationarity}We recall the idea of {\em mutual stationarity} introduced by Foreman and Magidor in \cite{FM2001} and a theorem due to Cummings, Foreman, and Magidor from \cite{CFM2006}.
\begin{defn} {\textbf{(cf. \cite[\textbf{Definition 6}]{FM2001} $\&$ \cite[\textbf{Definition 1.1}]{Apt2005})}
Let $\mathcal{K}$ be a set of regular cardinals with supremum $\lambda$. Suppose that $S_{\kappa}\subseteq \kappa$ for each $\kappa\in\mathcal{K}$. Then $\langle S_\kappa : \kappa\in \mathcal{K}\rangle$ is {\em mutually stationary} if and only if for all algebras $\mathcal{A}$ on $\lambda$, there is an elementary substructure $\mathcal{B}\prec\mathcal{A}$ such that for all $\kappa\in \mathcal{B}\cap \mathcal{K}$, sup($\mathcal{B}\cap\kappa)\in S_{\kappa}$.}
\end{defn}
\begin{thm} {\textbf{(Cummings, Foreman, and Magidor; cf. \cite[\textbf{Theorem 5.2}]{CFM2006})}}
{\em Let $\langle\kappa_{i}: i< \delta\rangle$ be an increasing sequence of measurable cardinals, where $\delta<\kappa_{0}$ is a regular cardinal. Let $S_{i}\subseteq \kappa_{i}$ be stationary for each $i<\delta$. Then $\langle S_i: i<\delta\rangle$ is mutually stationary.}
\end{thm}
\subsection{The Proof of Observation 1.11}
We recall the choiceless model constructed in \cite[\textbf{Theorem 1}]{Apt1983a} and the terminologies from \cite{Apt1983a}. In particular we fix an arbitrary $n_{0}\in \omega$ and assume an increasing sequence of measurable cardinals $\langle \chi_{k}:k<\omega\rangle$ in the ground model $V$ of $\mathsf{ZFC}$. Then we consider the choiceless model constructed in \cite[\textbf{Theorem 1}]{Apt1983a}. For the sake of convenience we call the model $\mathcal{N}_{n_{0}}$ and recall the relevant lemmas from \cite{Apt1983a}.
\begin{lem}{(cf. \cite[\textbf{Lemma 1.1}]{Apt1983a})}
{\em If $X\in \mathcal{N}_{n_{0}}$ is a set of ordinals, then $X\in V[G\restriction f]$ for some $f\in K$.}
\end{lem}
\begin{lem}{(cf. \cite[\textbf{Lemma 1.2}]{Apt1983a})}
{\em Let $\lambda=\bigcup_{k<\omega}\chi_{k}$. Then $\lambda=\aleph_{\omega}$ in $\mathcal{N}_{n_{0}}$
}
\end{lem}
\begin{proof}{\textbf{(of Observation 1.11)}} We note that in $\mathcal{N}_{n_{0}}$, $\chi_{k}=\aleph_{n_{0}+2(k+1)}$ for each $ k<\omega$. Let $\lambda=\bigcup_{k<\omega}\chi_{k}$ in $V$.
\begin{enumerate}
\item Following the arguments in \cite[\textbf{Lemma 1.36}]{Dim2011}, $\aleph_{n_{0}+2(k+1)}$ is a measurable cardinal in $\mathcal{N}_{n_{0}}$, for each $1\leq k<\omega$. Following \cite[\textbf{Theorem 4.3}]{HK2019}, for each $1\leq k<\omega$, there are no uniform ultrafilters on $\aleph_{n_{0}+2k+1}$ in $\mathcal{N}_{n_{0}}$. Consequently for each $1\leq k<\omega$, $\aleph_{n_{0}+2k+1}$ can not be a measurable cardinal in $\mathcal{N}_{n_{0}}$.
\item We prove that in the model $\mathcal{N}_{n_{0}}$, if $\langle S_{k}: 1\leq k <\omega\rangle$ is a sequence of stationary sets such that $S_{k}\subseteq\chi_{k}$ for every $1\leq k <\omega$, then $\langle S_{k}: 1\leq k <\omega\rangle$ is mutually stationary. By \textbf{Lemma 8.4}, $\lambda=\aleph_{\omega}$ in $\mathcal{N}_{n_{0}}$.
Suppose $\mathcal{N}_{n_{0}}\models ``\mathcal{A}$ is an algebra on $\lambda$ and $\langle S_{k}: 1\leq k <\omega\rangle$ is a sequence of stationary sets such that $S_{k}\subseteq\chi_{k}$ for every $1\leq k <\omega$''.
Since both $\mathcal{A}$ and $\langle S_{k}: 1\leq k <\omega\rangle$ can be coded by set of ordinals, by \textbf{Lemma 8.3}, there exists some $f\in K$ for which both $\langle S_{k}: 1\leq k <\omega\rangle \in V[G\restriction f]$ and $\mathcal{A}\in V[G\restriction f]$.
Following \cite[\textbf{Lemma 1.3}]{Apt1983a}, $\chi_k$ remains measurable in $V[G\restriction f]$ for every $1\leq k <\omega$. We can observe that $S_{k}$ is a stationary subset of $\chi_{k}$ in $V[G\restriction f]$ for every $1\leq k <\omega$. Fix any $1\leq k <\omega$. Let $C$ be any club subset of $\chi_{k}$ in $V[G\restriction f]$. Since the notion of club subset of $\chi_{k}$ is upward absolute and $V[G\restriction f]\subseteq \mathcal{N}_{n_{0}}$, $C$ is also a club subset of $\chi_{k}$ in $\mathcal{N}_{n_{0}}$. Since in $\mathcal{N}_{n_{0}}, S_{k}$ is a stationary subset of $\chi_{k}$ we have $S_{k}\cap C\not= \emptyset$. Thus, $S_{k}$ is a stationary subset of $\chi_{k}$ in $V[G\restriction f]$ for every $1\leq k <\omega$. By \textbf{Theorem 8.2}, $\langle S_{k}: 1\leq k <\omega\rangle$ is mutually stationary in $V[G\restriction f]$.
We note that $\mathcal{A}$ is an algebra on $\lambda$ in $V[G\restriction f]$.
Thus there is an elementary substructure $\mathcal{B}\prec\mathcal{A}$ in $V[G\restriction f]$ such that for all $k<\omega, sup(\mathcal{B}\cap\chi_{k})\in S_{k}$ by \textbf{Definition 8.1}. So there is an elementary substructure $\mathcal{B}\prec\mathcal{A}$ in $\mathcal{N}_{n_{0}}$ such that for all $k<\omega, sup(\mathcal{B}\cap\chi_{k})\in S_{k}$. Hence in $\mathcal{N}_{n_{0}}$, $\langle S_{k}: 1\leq k <\omega\rangle$ is mutually stationary.
\item We recall that $\lambda=\aleph_{\omega}$ in $\mathcal{N}_{n_{0}}$.
We can see that $\lambda$ is an almost Ramsey cardinal in $\mathcal{N}_{n_{0}}$ by a well-known argument from \cite[\textbf{Lemma 2.5}]{ADK2016}. For reader's convenience, we provide a sketch of the proof.
Let $f:[\lambda]^{<\omega}\rightarrow 2$ be in $\mathcal{N}_{n_{0}}$. Since $f$ can be coded by a set of ordinals, $f\in V[G\restriction f]$ for some $f\in K$ by \textbf{Lemma 8.3}. Following \cite[\textbf{Lemma 1.3}]{Apt1983a}, $\chi_k$ remains measurable in $V[G\restriction f]$ for every $1\leq k <\omega$. Consequently, $\chi_{k}$ is Ramsey in $V[G\restriction f]$ for every $1\leq k <\omega$. Now, in $V[G\restriction f]$, $\lambda$ is the supremum of Ramsey cardinals $\langle\chi_{i}: 1\leq i<\omega\rangle$.
Thus $\lambda$ is an almost Ramsey cardinal in $V[G\restriction f]$ by \cite[\textbf{Proposition 1}]{AK2008}.
Thus for all $\beta<\lambda$, there is a set $X_{\beta}\in V[G\restriction f]\subseteq \mathcal{N}_{n_{0}}$ which is homogeneous for $f$ and has order type at least $\beta$. Hence, $\lambda$ is almost Ramsey in $\mathcal{N}_{n_{0}}$ since $f$ was arbitrary.
\end{enumerate}
\end{proof}
\end{document} | math |
MP Board Exams 2022: मध्य प्रदेश बोर्ड ने 9वीं,11वीं कक्षा की डेटशीट की जारी, 15 मार्च से शुरू होंगे एग्जाम नई दिल्ली, एजुकेशन डेस्क। MP Board Exams 2022: मध्य प्रदेश बोर्ड ने 9वीं, 11वीं कक्षा की डेटशीट जारी कर दी है। मध्य प्रदेश बोर्ड ऑफ सेकेंड्री एजुकेशन Madhya Pradesh Board of Secondary Education, MPBSE ने 9वीं, 11वीं कक्षा की डेटशीट आधिकारिक वेबसाइट mpbse.nic.in पर रिलीज की है। शेड्यूल के अनुसार, 9वीं और 11वीं कक्षा के लिए एमपी बोर्ड परीक्षाएं 2022 15 मार्च और 16 मार्च, 2022 से शुरू होने वाली है। इसके साथ ही, कक्षा नवीं और ग्यारहवीं कक्षा की सभी परीक्षाएं सुबह 8:30 बजे से 11:30 बजे तक आयोजित की जाएंगी। सभी छात्रों को सुबह 8 बजे तक परीक्षा हॉल में उपस्थित रहने का निर्देश दिया गया है। सुबह 8:15 बजे के बाद किसी भी छात्र को परीक्षा केंद्र में प्रवेश की अनुमति नहीं दी जाएगी।छात्रों को परीक्षा के दौरान कोविड19 प्रोटोकॉल का सख्ती से पालन करना होगा। परीक्षा की अवधि में उस दौरान भी मास्क पहनना अनिवार्य होगा। इसके साथ ही एग्जाम में बैठने की व्यवस्था इस तरह से की जाएगी कि सोशल डिस्टेंसिंग का पालन किया जा सके। छात्रों को भी हमेशा अपने साथ हैंड सैनिटाइज़र रखना चाहिए।स्टूडेंट्स ध्यान दें कि अगर परीक्षा समय सारिणी के दौरान कोई सरकारी या सार्वजनिक अवकाश घोषित किया जाता है, तो भी परीक्षाएं यथावत जारी रहेंगी। सभी दिव्यांग छात्रों को 20 मिनट प्रति घंटे के हिसाब से अतिरिक्त समय और लेखन की सुविधा प्रदान की जाएगी। इसके अलावा, लेटेस्ट जानकारी के लिए, स्टूडेंट्स को सलाह दी जाती है कि वे mpbse.nic.in पर विजिट करते रहें। बोर्ड ने 9वीं, 11वीं की डेटशीट के अलावा, बोर्ड परीक्षाओं की तिथियों का भी ऐलान कर दिया है। यह परीक्षाएं 17 फरवरी से शुरू हो रही हैं। इसके तहत, दसवीं की बोर्ड परीक्षाएं 18 फरवरी से शुरू होकर 10 मार्च तक चलेंगी जबकि 12वीं बोर्ड की परीक्षाएं 17 फरवरी से शुरू होकर 12 मार्च को खत्म होंगी। बोर्ड परीक्षाओं की तैयारी माध्यमिक शिक्षा मंडल ने पूरी कर ली है। | hindi |
தினகரனை நம்பி போனவர்கள் நடுத்தெருவில் தான் நிற்க வேண்டும் முதலமைச்சர் பழனிசாமி விமர்சனம் அதிமுக பின்னடைவை சந்திக்க தினகரன் முயற்சி செய்வதாக முதலமைச்சர் பழனிசாமி தெரிவித்துள்ளார். சொத்துக் குவிப்பு வழக்கில் 4 ஆண்டுகள் பெங்களூரு பரப்பன அக்ரஹாராத்தில் உள்ள மத்திய சிறையில் அடைக்கப்பட்டு சிறைத்தண்டனை பெற்று வந்த சசிகலா கடந்த 27ஆம் தேதி விடுதலை செய்யப்பட்டார்.இதற்கு இடையில் கடந்த ஜனவரி 20ஆம் தேதி உடல்நலக்குறைவு காரணமாக மருத்துவமனையில் அனுமதிக்கப்பட்ட நிலையில் பின்பு அவருக்கு கொரோனா பாதிப்பு உறுதியான நிலையில் 11 நாட்கள் சிகிச்சைக்கு பிறகு டிஸ்சார்ஜ் செய்யப்பட்டார்.இதன் பின் நீண்ட நாட்கள் தனிப்படுத்தப்பட்டிருந்தார்.பின்பு நேற்று முன்தினம் பெங்களூருவில் இருந்து கிளம்பிய சசிகலா நேற்று சென்னை வந்தடைந்தார். நேற்று அமமுக பொதுச்செயலாளர் தினகரன் செய்தியாளர்களிடம் பேசினார்.அப்பொழுது அவர் கூறுகையில்,உறுப்பினர் அட்டை வழங்கும் அதிகாரமே பொதுச்செயலாளரிடம் தான் உள்ளது. புரட்சித்தலைவர் கண்ட அதிமுகவில் ஜெயலலிதா இருந்த வரையும் அதில் உள்ள சட்ட விதிகளில் பொதுச்செயலாளர் தான் எல்லா அதிகாரம் உடையவர். அவர் தான் பொதுக்குழுவை கூட்ட வேண்டும். பொதுச்செயலாளர் தான் ஒருவருக்கு பதவி நியமனம் செய்ய முடியும், பதவியில் இருந்து நீக்கமுடியும்.தற்போது அதை எல்லாம் நீக்கிவிட்டு உள்ளனர்.யாரையே இரண்டு பேரை போட்டுவிட்டு நடத்துவதற்கு இது என்ன கம்பெனியா ? என்றும் பொதுச்செயலாளருக்கான சட்டப்போராட்டத்தை சசிகலா தொடர்வார் என்றும் கூறினார். இந்நிலையில் நேற்று வேலூரில் முதலமைச்சர் பழனிசாமி தேர்தல் பிரச்சாரத்தில் ஈடுபட்டார்.அப்பொழுது அவர் பேசுகையில், அதிமுக பின்னடைவை சந்திக்க சில பேர் முயற்சி செய்கின்றனர். இதில் ஒருவர் தினகரன்.10 ஆண்டுகளாக கட்சியிலே கிடையாது.ஜெயலலிதா அடிப்படை உறுப்பினர் பதவியில் இருந்து தினகரனை நீக்கி வைத்திருந்தார். ஜெயலலிதா மறைவிற்கு பிறகு ஒரே கட்சியில் இணைந்து கொண்டதாக அறிவித்துக்கொண்டார்.அதிமுகவை கைப்பற்ற தினகரன் பல்வேறு முயற்சிகளை மேற்கொண்டார்.எங்களது எம்எல்ஏக்கள் 18 பேரை பிடித்து வைத்துக்கொண்டார்.18 பேரையும் நடுரோட்டில் விட்டுவிட்டார். தினகரனை நம்பி போனவர்கள் நடுத்தெருவில் தான் நிற்க வேண்டும் என்று பேசியுள்ளார். | tamil |
You know if its on Brainfeeder its gonna be funky, and Louis Cole doesnt disappoint. Catch him performing a solo show at the Festsaal Kreuzberg this May.
Louis’s new album features his friend Thundercat (with whom he worked on “Drunk” co-writing ‘Bus In The Streets’ and ‘Jameel’s Space Ride’) who returns the favour, contributing lead vocals on ‘Tunnels In The Air’. Genevieve Artadi and acclaimed jazz pianist and experimental composer Brad Mehldau also pop up on ‘When You’re Ugly’ and ‘Real Life’ respectively. Furthermore, ‘Last Time You Went Away’ features a 23-piece string orchestra – the Rochester Stringz from Eastman School of Music. | english |
Belur Math : ఈ నెల 18న తెరచుకోనున్న బేలూరు మఠం కోల్కతా : రామకృష్ణ మిషన్ ప్రధాన కార్యాలయమైన బేలూరు మఠంలోకి సందర్శకులను ఈ నెల 18 నుంచి అనుమతించనున్నారు. ఈ సందర్భంగా సందర్శకులు తప్పనిసరిగా కొవిడ్ నిబంధనలు పాటించాలని పరిపాలన స్పష్టం చేసింది. ప్రతి రోజు ఉదయం 8 నుంచి 11 గంటల వరకు, సాయంత్రం 4 గంటల నుంచి 5.45గంటల వరకు తెరిచి ఉంటుందని తెలిపింది. సందర్శకులు తప్పనిసరిగా మాస్క్ ధరించాలని, చేతులు శానిటైజర్తో శుభ్రం చేసుకోవాలని సూచించింది. మఠం ప్రాంగణంలోకి ప్రవేశించే ముందు థర్మన్ స్క్రీనింగ్ పరీక్షలు చేయనున్నట్లు పేర్కొంది. సందర్శకులు వ్యాక్సినేషన్ సర్టిఫికెట్, 72 గంటల్లో తీసుకున్న ఆర్టీపీసీఆర్ పరీక్ష నెగెటివ్ కాపీ, ఆధార్, పాన్, ఓటరు ఐడీ వంటి గుర్తింపు కార్డులను చూపిస్తేనే మఠంలోకి అనుమతి ఇవ్వనున్నట్లు పేర్కొంది.అన్ని కోవిడ్ అప్డేట్స్ గురించి తెలుసుకునేందుకు ఇక్కడ చదవండి కరోనా మహమ్మారి రెండో దశవ్యాప్తి నేపథ్యంలో మఠాన్ని మూసివేయగా.. ఇటీవల గురు పౌర్ణమి సందర్భంగా జూలై 24న మఠాన్ని ఒక రోజు తెరిచారు. ఇవి కూడా చదవండి.. GSLVF10 : జీఎస్ఎల్వీ రాకెట్ కౌంట్డౌన్ ప్రారంభం Algeria : చెలరేగిన మంటలు.. 25 మంది సైనికుల మృత్యువాత శ్రీశైలం, సాగర్కు కొనసాగుతున్న ప్రవాహం | telegu |
Jacqueline Fernandez: జాక్వెలిన్ ఫెర్నాండేజ్ కేక పెట్టిస్తుందిగా.. Jacqueline Fernandez: శ్రీలంకన్ బ్యూటీ జాక్వెలిన్ ఫెర్నాండేజ్ తన అందచందాలతో వెండితెరపై ఎంతగా అలరించిందో ప్రత్యేకంగా చెప్పనక్కర్లేదు. ఈ అమ్మడు ఎప్పటికప్పుడు కొత్తగా కనిపిస్తూ అందరి అటెన్షన్ తన వైపుకు తిప్పుకుంటుంది. కరోనా సమయంలో సల్మాన్ ఫాం హౌజ్ లో ఉన్న జాక్వెలిన్ తెగ సందడి చేసింది. జాక్వెలిన్ అప్పుడప్పుడు హాట్ ఫొటోలు షేర్ చేస్తూ.. హీటెక్కిస్తుంటుంది. తాజాగా ఈ అమ్మడు జాక్వెలిన్ స్టన్నింగ్ లుక్ కి సంబంధించిన ఫొటోలు సోషల్ మీడియాలో షేర్ చేస్తూ కేక The post Jacqueline Fernandez: జాక్వెలిన్ ఫెర్నాండేజ్ కేక పెట్టిస్తుందిగా.. first appeared on The News Qube. Jacqueline Fernandez: శ్రీలంకన్ బ్యూటీ జాక్వెలిన్ ఫెర్నాండేజ్ తన అందచందాలతో వెండితెరపై ఎంతగా అలరించిందో ప్రత్యేకంగా చెప్పనక్కర్లేదు. ఈ అమ్మడు ఎప్పటికప్పుడు కొత్తగా కనిపిస్తూ అందరి అటెన్షన్ తన వైపుకు తిప్పుకుంటుంది. కరోనా సమయంలో సల్మాన్ ఫాం హౌజ్ లో ఉన్న జాక్వెలిన్ తెగ సందడి చేసింది. Jacqueline Fernandez Stunning Photos జాక్వెలిన్ అప్పుడప్పుడు హాట్ ఫొటోలు షేర్ చేస్తూ.. హీటెక్కిస్తుంటుంది. తాజాగా ఈ అమ్మడు జాక్వెలిన్ స్టన్నింగ్ లుక్ కి సంబంధించిన ఫొటోలు సోషల్ మీడియాలో షేర్ చేస్తూ కేక పెట్టిస్తుంది. తన ప్రైవేట్ పార్ట్స్ కి వేలెట్ అడ్డు పెట్టి దిగిన పిక్ వావ్ అనిపిస్తుంది. Jacqueline Fernandez Stunning Photos బాలీవుడ్ నటి జాక్వెలిన్ ఫెర్నాండెజ్ కు డ్రగ్స్, మనీలాండరింగ్ కేసుల్లో భాగంగా ఇటీవల సమన్లు జారీ చేశారు ఈడీ అధికారులు. అంతేకాదు సెప్టెంబర్ 25 వ తేదీన విచారణకు హాజరుకావాలని ఈడీ అధికారులు స్పష్టం చేశారు. చీటర్ సురేష్ తో సంబంధాలపై జాక్వెలిన్ ను ఎన్ఫోర్స్మెంట్ డైరెక్టరేట్ అధికారులు విచారించనున్నట్లు సమాచారం. నటి జాక్వెలిన్ ఫెర్నాండెజ్ ను ఈడీ అధికారులు ఇదివరకే ప్రశ్నించగా.. మరోసారి విచారణకు హాజరు కావాలంటూ సమన్లు జారీ చేయడంతో బాలీవుడ్ లో హాట్ టాపిక్ గా మారింది. Jacqueline Fernandez Stunning PhotosThe post Jacqueline Fernandez: జాక్వెలిన్ ఫెర్నాండేజ్ కేక పెట్టిస్తుందిగా.. first appeared on The News Qube. | telegu |
ગ્રે વ્હેલને કિસ કરી તો તેણે પાણીની છોળો ઉડાડી મેક્સિકોના માગદાલેના ખાડીના પ્રવાસે નીકળેલી ૩૬ વર્ષની ઍલેક્સ બૅન્કીએ તેની બોટની નજીક ૪૫ ટન વજનની ગ્રે વ્હેલને સપાટી પર આવતી જોઈ હતી, જે પાણીમાં ઊંડે જતાં પહેલાં બોટની નજીક આવીને ટૂરિસ્ટ્સને મોહિત કરે છે. પહેલી ફેબ્રુઆરીએ લેવામાં આવેલા આ વિડિયોમાં ગ્રે વ્હેલ બોટની એટલી નજીક આવે છે કે ટૂરિસ્ટો એને કિસ કરી શકે છે. ઑનલાઇન પોસ્ટ કરાયેલો આ વિડિયો અત્યાર સુધી ૨,૪૮,૦૦૦ વાર જોવાયો છે, જેમાં ગ્રે વ્હેલ બૅન્કી અને અન્ય મુસાફરોની નજીક પોતાનું માથું લઈ જઈને તેમને નિકટતાનો અનુભવ કરાવ્યા બાદ એની પૂંછડીથી તેમના પર પાણીની છાલક મારે છે. બૅન્કી જણાવે છે કે આ પહેલાં પણ તેણે વ્હેલ માછલી જોઈ છે, પણ આટલી નજીકથી સ્પર્શ કરી શકાય, કિસ કરી શકાય એ રીતે નજીક નહોતી આવી. લગભગ ૪૦થી ૫૦ ફીટ લાંબી આ વ્હેલ સ્વચ્છ, સુંવાળી, રબર જેવી મુલાયમ અને દરિયાની સુવાસથી ભરેલી હતી. પોતાની પૂંછડીની એક હળવી થપાટથી આખી બોટને પલટાવવાની ક્ષમતા ધરાવતી ગ્રે વ્હેલ બોટ પરના પર્યટકોને ઈજા પહોંચાડવાને બદલે ગેલ કરાવે છે એ જ સૂચવે છે કે ઈશ્વરે પ્રત્યેક જીવમાં દયામાયા મૂકી છે. | gujurati |
బిగ్ బాస్ 5 : వాళ్ళను గుర్తు చేస్తున్నావ్. సిరిపై ట్రోలింగ్ బిగ్ బాస్ సీజన్ 5 కంటెస్టెంట్ సిరిని నెటిజన్లు దారుణంగా ట్రోల్ చేస్తున్నారు. టాస్క్ సమయంలో సన్నీ తనను అసభ్యకరంగా తాకాడంటూ సిరి ఆరోపించడంతో దానికి కారణమైంది. బిగ్ బాస్ సీజన్ 2లో భాను, తేజస్విలతో ఆమెను పోలుస్తున్నారు. సీజన్ 2 సమయంలో భాను శ్రీ, తేజస్వి మరో కంటెస్టెంట్ కౌశల్తో ఇలాగే ప్రవర్తించారు. ఫిజికల్ టాస్క్ సమయంలో లేడీ కంటెస్టెంట్లను అతను అసభ్యకరంగా తాకాడంటూ ఆరోపించారు. అప్పట్లో ఈ విషయం సంచలనం సృష్టించింది. వారు ఆరోపించినట్లుగా కౌశల్ తాకలేదని తరువాత ప్రూవ్ అవ్వడంతో నెటిజన్లు లేడీ కంటెస్టెంట్లపై విరుచుకు పడ్డారు. కావాలనే కౌశల్ ను టార్గెట్ చేశారంటూ వాళ్ళను ఎలిమినేట్ చేసేదాకా శాంతించలేదు. Read Also : బిగ్ బాస్ హౌస్ లో ఆ గట్టునుంటావా. ఈ గట్టునుంటావా! అయితే అప్పట్లో భాను విషయంలో కౌశల్ నిజంగానే అలా చేశాడా ? సందేహం వచ్చింది ప్రేక్షకులకు. అందుకే ముందుగా ప్రేక్షకులు కొంతమంది కౌశల్ కు సపోర్ట్ చేస్తే, మరికొంత మంది మాత్రం భానుకు సపోర్ట్ చేశారు. కానీ ఇప్పుడు సిరి విషయంలో సన్నీ తప్పు చేయలేదని చాలా స్పష్టంగా ఉంది. అతను సిరి నుండి లాఠీలను దొంగిలించడానికి శ్వేత సహాయం తీసుకున్నాడు. ఇంతకుముందు వరకు లోబో సిరిని సీతాకోకచిలుక పిలవడంతో సిరిపై ప్రేక్షకులకు మంచి అభిప్రాయం ఏర్పడింది. కానీ ఇప్పుడు అదే నెటిజన్లు ఆమెను ట్రోల్ చేస్తున్నారు. ఆమె చీప్ ట్రిక్స్ తో సానుభూతి పొందడానికి ప్రయత్నిస్తోందని అంటున్నారు. మొత్తం మీద ఆమె సన్నీ మీద చేసిన ఆరోపణలు బెడిసి కొట్టాయి. అతనికి సానుభూతి పెరుగుతుండగా, సిరిపై మాత్రం నెగెటివిటీ ఎక్కువయ్యింది. | telegu |
Atlantis grows two varieties of personal size baby watermelons, The Black Beauty and the Baby Gold! Both of these watermelons are succulent and sweet. Atlantis picks them at the peak of sweetness; straight from the field to your customers! Melons range in size from 2 - 3.5 lbs. | english |
ভিয়েতনামের সাথে বিশেষ বৈঠক মোদীর! কি হল সেই বৈঠকে ভিয়েতনামের কমিউনিস্ট পার্টির সাধারণ সম্পাদক নগুয়েন ফু ট্রংয়ের সাথে বিশেষ বৈঠক বসলেন প্রধানমন্ত্রী নরেন্দ্র মোদী শুক্রবার ইউক্রেনের সঙ্কট এবং দক্ষিণ চীন সাগরের পরিস্থিতি নিয়ে পর্যালোচনা করতে টেলিফোনিক বৈঠকের আয়োজন করা হয় দুই দেশের তরফে এই বৈঠকে নরেন্দ্র মোদী ও নগুয়েন ফু ট্রং ভারত এবং ভিয়েতনামের ব্যাপক কৌশলগত অংশীদারিত্বে সহযোগিতার ক্ষেত্রগুলির বিষয়ে কথা বলেন যা ২০১৬ সালে মোদীর ভিয়েতনাম সফরের সময় প্রতিষ্ঠিত হয় এছাড়াও দুই দেশের মধ্যে কূটনৈতিক সম্পর্ক প্রতিষ্ঠার ৫০তম বার্ষিকী উপলক্ষে একে অপরকে অভিবাদন জানান বলেও জানা যায় পাশাপাশি দুই দেশের সামরিক ক্ষেত্রে পারস্পরিক সমঝোতা বৃদ্ধির বিষয়ে বিশেষ নজর দেওয়ার ব্যাপারেও আগ্রহী হয়েছেন এদেশের কেন্দ্রের তরফে বিবৃতিতে বলা হয়েছে, ভারতের ফার্মা এবং কৃষিপণ্যের জন্য ভিয়েতনামের বাজারে বেশি সুযোগ এবং সুবিধা দেওয়ার জন্য অনুরোধ করেছেন মোদী এখন সেই বিষয়ে ভিয়েতনামের কতটা সম্মতি পাওয়া যায় তার অপেক্ষায় ভারত এছাড়াও এই বৈঠকে দুই দেশের বর্তমান উদ্যোগগুলির ক্ষেত্রে দ্রুত অগ্রগতির জন্য কাজ করার পাশাপাশি দ্বিপাক্ষিক সম্পর্কের সুযোগ বাড়ানোর বিষয়ে কথা বলেছেন তারা আর ও পড়ুন বিভূতিভূষণ বন্দোপাধ্যায়ের প্রাণের ইছামতীর পাড়ে নববর্ষ পালন উল্লেখ্য, ভিয়েতনামের কমিউনিস্ট পার্টির সাধারণ সম্পাদক নগুয়েন ফু ট্রংয়ের সাথে বিশেষ বৈঠক বসলেন প্রধানমন্ত্রী নরেন্দ্র মোদী শুক্রবার ইউক্রেনের সঙ্কট এবং দক্ষিণ চীন সাগরের পরিস্থিতি নিয়ে পর্যালোচনা করতে টেলিফোনিক বৈঠকের আয়োজন করা হয় দুই দেশের তরফে | bengali |
Certified by USA Triathlon as a Level 1 Coach!
I have 5 years experience racing triathlon and 3 of them professionally. My first race was in 2010 (so yes I still remember those first race butterflies! ) and I qualified for my pro card the weekend of my 21st Birthday in 2012. My first season pro was in 2013 and last year I had a phenomenally successful season. My high school background in swimming and running, as well as my opportunity to work with various coaches has given me a vary extensive background and knowledge in the sport.
Whether you are a beginner getting ready for your first race or someone looking to set PR’s this season, I would love to help you achieve your goals.
Coaching can be done remotely from anywhere in the world! I use the training peaks system to track and analyze all of my athletes training. | english |
دٲمِس عالمی جنگہٕ پتہٕ تام تہٕ رودٕ استاد بحثس پیٹھ حٲوی۔ | kashmiri |
એક્શન ભારતના તમામ નાગરિકોને હેમખેમ આવશે સ્વદેશ, મોદી સરકારે જુઓ શું બનાવ્યો માસ્ટર પ્લાન યુક્રેનમાં ફસાયેલા ભારતીયોની વહારે આવી મોદી સરકાર તમામ નાગરિકો આવશે હેમખેમ 2 દેશોની મદદથી શરૂ કર્યું મોટું અભિયાન સરકારે યુક્રેનના પશ્ચિમી શહેરોમાં બે કેમ્પ ઓફિસ શરૂ કરી છે, જેના દ્વારા યુક્રેનથી ત્યાં ભણતા વિદ્યાર્થીઓની પ્રથમ બેચ Indian Students Trapped in Ukraine લાવવાની કામગીરી શરૂ કરવામાં આવી છે.Click here to get the latest updates on Ukraine Russia conflict યુક્રેનના 2 શહેરોમાં શિબિર કચેરીઓ સ્થપાઈ સૂત્રોના જણાવ્યા અનુસાર, વિદેશ મંત્રાલયે યુક્રેનના પશ્ચિમી શહેરો Lviv અને Chernivtsiમાં બે કેમ્પ ઓફિસ શરૂ કરી છે. આ બંને કેમ્પમાં રશિયન અને યુક્રેનિયન બોલતા અધિકારીઓને તૈનાત કરવામાં આવી રહ્યા છે. રોમાનિયા થઈને ભારત લાવવાનો પ્રયાસ કરી રહ્યા છે યુક્રેનના અલગઅલગ ભાગોમાં રહેતા, મેડિકલ અને અન્ય કોર્સનો અભ્યાસ કરી રહેલા આ અધિકારીઓ ભારતીય વિદ્યાર્થીઓને ત્યાંથી બહાર કાઢીને બસ દ્વારા આ બે શહેરોમાં લઈ જશે. ત્યારબાદ, રોડ મારફતે બોર્ડર ક્રોસ કરીને, આ વિદ્યાર્થીઓને રોમાનિયાથી એર ઈન્ડિયાની વિશેષ ફ્લાઈટ દ્વારા ભારત પરત લાવવામાં આવશે. પ્રથમ બેચને બહાર કાઢવાનું કામ શરૂ કર્યું ભારતે યુક્રેનમાં ફસાયેલા વિદ્યાર્થીઓIndian Students Trapped in Ukraineની પ્રથમ બેચને બહાર કાઢવાનું કામ શરૂ કરી દીધું છે. યુક્રેનથી વિદ્યાર્થીઓની પ્રથમ બેચ Chernivtsi શહેરમાં લાવવામાં આવી હતી. ત્યાંથી તેને બસ મારફતે રોમાનિયા લઈ જવામાં આવ્યા, જ્યાંથી તેઓ ફ્લાઈટ મારફતે ભારત આવશે. યુક્રેનમાં 20 હજાર ભારતીયો રહે છે જણાવી દઈએ કે યુક્રેનમાં ભારતના લગભગ 18 હજાર વિદ્યાર્થીઓ અને 2 હજાર સામાન્ય લોકો રહે છે. રશિયા અને યુક્રેન વચ્ચે યુદ્ધ શરૂ થયા બાદ ત્યાંની સ્થિતિ ઝડપથી બગડી રહી છે અને તેઓ ભારત સરકારને બચાવ અભિયાન શરૂ કરવાની વિનંતી કરી રહ્યા છે. મોદી સરકારે શુક્રવારે જાહેરાત કરી હતી કે તે યુક્રેનમાં રહેલા તમામ ભારતીયોને પોતાના ખર્ચે સુરક્ષિત રીતે પરત લાવશે અને ભારત પરત લાવશે. આ માટે ઝુંબેશ શરૂ કરવામાં આવી છે. | gujurati |
/*
* Copyright (c) 2008-2016 Haulmont.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package com.haulmont.restapi.service;
import com.google.common.base.Joiner;
import com.haulmont.chile.core.model.MetaClass;
import com.haulmont.cuba.core.app.serialization.ViewSerializationAPI;
import com.haulmont.cuba.core.global.Messages;
import com.haulmont.cuba.core.global.Metadata;
import com.haulmont.cuba.core.global.View;
import com.haulmont.cuba.core.global.ViewRepository;
import com.haulmont.restapi.common.RestControllerUtils;
import com.haulmont.restapi.data.MetaClassInfo;
import com.haulmont.restapi.exception.RestAPIException;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import org.springframework.http.HttpStatus;
import org.springframework.stereotype.Component;
import javax.inject.Inject;
import java.util.*;
import java.util.stream.Collectors;
/**
* Class is used by the {@link com.haulmont.restapi.controllers.EntitiesMetadataController}. Class is sed for getting
* entities metadata. User permissions for entities access aren't taken into account at the moment.
*/
@Component("cuba_EntitiesMetadataControllerManager")
public class EntitiesMetadataControllerManager {
private static final Logger log = LoggerFactory.getLogger(EntitiesMetadataControllerManager.class);
@Inject
protected Metadata metadata;
@Inject
protected RestControllerUtils restControllersUtils;
@Inject
protected ViewSerializationAPI viewSerializationAPI;
@Inject
protected ViewRepository viewRepository;
@Inject
protected Messages messages;
public MetaClassInfo getMetaClassInfo(String entityName) {
MetaClass metaClass = restControllersUtils.getMetaClass(entityName);
return new MetaClassInfo(metaClass);
}
public Collection<MetaClassInfo> getAllMetaClassesInfo() {
Set<MetaClass> metaClasses = new HashSet<>(metadata.getTools().getAllPersistentMetaClasses());
metaClasses.addAll(metadata.getTools().getAllEmbeddableMetaClasses());
return metaClasses.stream()
.filter(metaClass -> metadata.getExtendedEntities().getExtendedClass(metaClass) == null)
.map(MetaClassInfo::new)
.collect(Collectors.toList());
}
public String getView(String entityName, String viewName) {
MetaClass metaClass = restControllersUtils.getMetaClass(entityName);
View view = viewRepository.findView(metaClass, viewName);
if (view == null) {
throw new RestAPIException("View not found",
String.format("View %s for metaClass %s not found", viewName, entityName),
HttpStatus.NOT_FOUND);
}
return viewSerializationAPI.toJson(view);
}
public String getAllViewsForMetaClass(String entityName) {
MetaClass metaClass = restControllersUtils.getMetaClass(entityName);
StringBuilder sb = new StringBuilder();
sb.append("[");
List<String> jsonViews = new ArrayList<>();
for (String viewName : viewRepository.getViewNames(metaClass)) {
View view = viewRepository.getView(metaClass, viewName);
jsonViews.add(viewSerializationAPI.toJson(view));
}
sb.append(Joiner.on(",").join(jsonViews));
sb.append("]");
return sb.toString();
}
}
| code |
Amnesty International (AI) and the Canadian Council for Refugees (CCR) released a brief today calling for Canada to suspend the Safe Third Country Agreement with the United States.
The 52-page brief, Contesting the Designation of the US as a Safe Third Country, outlines the many ways that the US asylum system and immigration detention regime fail to meet required international and Canadian legal standards. It highlights how law and practice have deteriorated further since President Donald Trump took office.
Despite many calls from refugee and human rights organizations and legal academics on both sides of the border following President Trump’s issuance of Executive Orders earlier this year, the Canadian government has repeatedly stated that there is no need to revisit the Agreement. Minister of Immigration, Refugees and Citizenship Ahmed Hussen has maintained that position following his review of the AI/CCR brief.
Under the Safe Third Country Agreement refugees who present themselves at a Canada-US border post seeking to make a refugee claim in Canada are, with limited exceptions, denied access to the Canadian refugee system and immediately returned to the United States. The Agreement does not apply to individuals who cross irregularly into Canada, other than at a border post, and subsequently make a refugee claim inside Canada. Those making irregular border crossings from the US face perilous situations in inclement weather and isolated locations, as well as exploitation by people smugglers.
One-Year Bar – With limited exceptions, asylum seekers cannot make a claim if they have been within the United States for more than a year. The bar has a disproportionately harsh impact on certain refugees, including women and LGBTI claimants.
Expansion of Expedited Removal – these proceedings permit the removal of certain groups of non-citizens from the United States without a hearing before an Immigration Judge.
Detaining Asylum Seekers – detention is used unlawfully as a punitive and arbitrary measure; conditions and location of detention impede access to legal counsel. Serious deficiencies in detention conditions include inadequate access to medical care, prolonged confinement in holding cells, and prison like conditions with severe psychological impacts. New policies will substantially increase the use of immigration detention and exacerbate these problems.
Operation Streamline and Prosecution of Asylum Seekers – Contrary to international law, asylum-seekers face the risk of prosecution for “illegal” entry into the United States.
Inconsistent Recognition of Gender-Based Asylum Claims – There is an inconsistent record of recognizing claims of gender-based persecution.
Inconsistent Adjudication – rates of acceptance of similarly situated asylum claims vary dramatically between different regions. Some areas of the United States with exceedingly low acceptance rates are effectively “asylum free zones”.
To add your voice, participate in the online action here. | english |
கால சர்ப்பதோஷம் தோஷம் என்றால் சமஸ்கிருத மொழியில் குறைபாடு அல்லது இழப்பு என்று பெயர். ஆதாவது ஜெனன கால சக்கரத்தில் அனைத்து கிரகங்களும், ராகு மற்றும் கேது என்ற இரண்டு சர்பங்களின் பிடியில் மாட்டி கொள்வது கால சர்ப்ப தோசம் என்று ஜோதிடம் கூறுகிறது. இதன் சூட்சுமம் என்னவென்றால், ராகு மற்றும் கேது என்பது நம் முன்னோர்கள் வழிவழியாய் தொடர்ந்து செய்து வரும் பாவங்களை குறிக்கும் ஒரு குறியீடு என்பதே . ராகு என்பது அதிவேகமாக பரவி செல்வது அல்லது கட்டுபாடில்லா வேகம் என்றே சமஸ்கிருதத்தில் பொருள். அடக்கமுடியாத போகத்தின் உருவாக வரும் ராகு நம் முன்னோர்களின் ஆதாவது தந்தை வழி தந்தை செய்த இயற்கைக்கு புறம்பான பாவங்களையும், கேது தாய் வழி முன்னோர்கள் செய்த இயற்கைக்கு புறம்பான பாவங்களையும் காட்டுகிறது. இதில் இன்னொரு சூட்சுமம், கேது சந்திரனின் பிம்பம் என்பதால், தாய் வழி பாவத்தை குறிக்கிறார். ராகு சூரியனை பாதிப்பதால், தந்தை வழி முன்னோர்களின் பாவங்கள் என்பதை அறியலாம். தற்கால அறிவியல் படி கூறவேண்டுமென்றால், மனித ஜீனில் அடிப்படைகளாக இருக்கும் 46 குரோமோசோம்கள் ராகு கேது கலவைகள் ஆகும். குரோமோசோம்கள் என்பவை ஒருவரின் உயரம், தோல், முடி, கண்விழி ஆகியவற்றின் நிறம், அறிவுத்திறன், பேசும் விதம், முகத்தோற்றம், உடல்பருமன், பரம்பரைவியாதி இதுபோன்ற அனைத்து குணங்களும் பதிவாகி உள்ளது. ஆதாவது 23 குரோமோசோம்கள் தந்தை வழி குறிப்பவை ராகு மற்றும் 23 குரோமோசோம்கள் தாய் வழி குறிப்பவை கேது. இதில் இருந்து நம் ஜோதிட ரிஷிகள் எந்த அளவு அறிவியல் திறன் பெற்றவர்கள் என அறியலாம். ராகுவும் கேதுவும் வட மற்றும் தென் துருவங்கள் என்று ஜோதிடம் கூறுகிறது. இரண்டும் ஒன்றுகொன்று எதிரெதிர் குணங்களை கொண்டவைகள். அவ்வாறு இரண்டு வெவ்வேறு குணாதிசயம் பெற்ற இரு உருவமற்ற சாயா கிரகங்களிடையே மாட்டும் மற்ற கிரகங்களின் நிலைமை நாம் சொல்லி தெரிய வேண்டியது இல்லை. ராகு கருநாக பாம்பாகவும், கேது செம்பாம்பாகவும் ஜோதிடம் கூறுகிறது. சுருங்க கூறினால் மனதின் சந்திரன் ஆசைகளை களையும் நிழல் கேது. பரமாத்மாவை சூரியனை கெடுக்கும் நிழல் ராகு. இவ்வாறு, மனித உடலில் இருக்கும் ஜீன்களின் நம் பரம்பரை தன்மை குறிப்பவை ராகு மற்றும் கேது ஆகும். பாவங்களை சுமக்கும் ராகு கேது என்னும் இரு சர்பங்கள் மத்தியில் மாட்டும் அனைத்து கிரகங்களும் பலம் இழக்கிறது. அதனால், இவ்வளவு பெரிய யோகங்கள் ஜாதகத்தில் இருந்தாலும் அந்த யோகம் தரும் கிரகங்களின் திசை புத்திகள் ஜாதகரின் வாழ்வில் பெரிய மாற்றத்தை கொடுப்பதில்லை. ஒரு ராசியில் 1.5 ஆண்டுகள் இருந்து, 12 ராசிகளை கடக்க 12 X 1.5 18 அவை 18 ஆண்டுகள். அவ்வாறு ராகு 18 ஆண்டுகள் மற்றும் கேது 18 ஆண்டுகள். ஜாதக கால சக்கரத்தை கடக்க 36 ஆண்டுகள் எடுத்து கொண்டு முழுமையாக தன் சுற்றை முடித்து தோஷ நிவர்த்தி தருகிறது. 36 ஆண்டுகள் கழித்து, ஜாதகத்தில் உள்ள யோக பலன்களை தருகிறது. திருமண தடை, வேலையின்மை, நோய் எதிப்பு சக்தி குறைதல், அதிஷ்டம் இன்மை, குடும்பத்தில் குழப்பம், உறவினர்களுடன் ஒற்றுமை இன்மை என பல மன நிமைதியை இழக்கும் நிகழ்வுகளை தருகிறது. எனவே ஜாதக பலன்களை தகுந்த நேரத்தில் தாராமல், சர்ப்பம் என்னும் ராகு கேதுக்களால் காலம் தாழ்த்தி தரபடுவதால், இது கால சர்ப்ப தோஷம் என்று பெயர் பெற்றது. ஆதாவது முன்னோர்கள் செய்த பாவங்களை ராகு என்னும் பாம்பின் வாயின் பிடியில் சிக்கி வேதனை பட்டு 36 ஆண்டுகள் கழித்து, கேது என்னும் பாம்பின் பல அனுபவங்கள் பெற்று வால் பகுதியில் தோஷ நிவர்த்தி கிடைக்கிறது. உண்மையில் கால சர்ப்ப தோசம் என்பது பாம்புகளால் உருவாக்கவில்லை. ஜோதிடத்தில் அனைத்தும் ஒரு வகை குறியீடுகளே ராகு கேது என்பது கொடிய பலன்களை தரும் கிரகங்கள் என்பதால் அவை சர்ப்பன்களுக்கு ஒப்பிடு செய்யபடுகின்றன. இவை உங்கள் மூதாதையர் செய்த பாவங்களே. அவற்றை போக்க இறை வழிபடு, யோக சித்தி அடைந்த சித்தர்கள், மகான்களின் ஜீவ சமாதி வழிபாடு,தான தர்மங்கள், குல தெய்வ வழிபாடு, நம் முன்னோர்களின் காலில் விழுந்து ஆசி பெறுதல் இவை எல்லாம் ராகு கேதுவால் உருவாகும் கெடு பலன்களை தாங்கும் சக்தியை நம் உடலுக்கு மனதுக்கும் அளிக்கிறது. முடிவாக, ராகுவும் கேதுவும் தரும் அனைத்து தோஷங்களும் முன்னோர்கள் வழி வந்தவை என்பதை மனதில் கொள்க. அதனை முறையாக இறைவழிபாடு மூலமும், சித்தர்கள் அருள் பெற்றும் தோச நிவர்த்தி அடையலாம். நிகழ காலத்தில் செய்யும் நல்ல செயல்கள், தோச நிவர்த்திக்கு செய்யும் பிராத்தனைக்கு உறுதுணையாக இருக்கும் என்பது என் நம்பிக்கை. கால சர்ப்ப தோசம் கூறும் விடயம் என்னவெனில், தற்காலத்தில் நாம் செய்யும் பாவம், நம் சந்ததிகளை ராகு மற்றும் கேது உருவில் பாதிக்கும் என்பதே. | tamil |
لاہور نمائندہ سپورٹس پاکستانی کرکٹ ٹیم کے ال راونڈر عماد وسیم نے دعوی کیا ہے کہ جب وہ بائولنگ کرتے ہیں تو خود کو مکمل بائولر اور جب بیٹنگ کرتے ہیں تو خود کو ایک بلے باز کے طور پر لیتے ہیں میں سمجھتا ہوں کہ میں ایک ال راونڈر ہوں جو بیٹنگ اور بائولنگ دونوں سے پاکستان کو میچز جتوا سکتا ہوں کارکردگی میں عدم تسلسل پر تنقید کے حوالے سے سوال کیا گیا تو ان کا کہنا تھا کہ ناقدین کا کام تنقید کرنا ہے میں نے ہمیشہ اپنی سو فیصد کارکردگی دکھانے کی کوشش کی ہے اور ائندہ بھی ایسا ہی کروں گاانہوں نے خود کو اسٹار ال راونڈر بوم بوم شاہد خان افریدی کا متبادل قرار دینے سے انکار کیا اور کہا کہ شاہد افریدی ایک سپر سٹار ہیں اور انہوں نے جو پاکستان کیلئے پرفارم کیا اس کا میں سوچ بھی نہیں سکتا اور نہ ہی ان کی برابری کر سکتا ہوںعماد وسیم نے دورہ جنوبی افریقہ میں اچھی کارکردگی کے عزم کا اظہار بھی کیا | urdu |
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title>Title</title>
</head>
<body>
</body>
<script>
var stringValue = "chinese is a beautiful country！"
var positions = new Array();
var pos = stringValue.indexOf("e")
while (pos > -1){
positions.push(pos);
pos = stringValue.indexOf("e",pos+1);
}
alert(positions);
</script>
</html> | code |
शतप्रतिशत टीकाकरण करने पर किया सम्मान जागरण संवाददाता, ज्ञानपुर भदोही : किशोरों को कोरोना संक्रमण से बचाने के लिए चलाए जा रहे अभियान के तहत आठ विद्यालयों में शतप्रतिशत छात्रछात्राओं का टीकाकरण सुनिश्चित कराने पर सभी के प्रधानाचार्यों को सोमवार को सम्मानित किया गया। विद्यालयों में टीकाकरण के लिए नोडल अधिकारी बनाए गए उप कृषि निदेशक अरविंद कुमार सिंह ने माल्यार्पण कर सम्मानित किया तो अन्य शिक्षक व कर्मचारियों सहित स्वास्थ्य टीम के प्रयास की सराहना की। उन्होंने बताया कि लक्ष्मण प्रसाद इंटर कालेज करियांव 15 से 18 वर्ष के सभी 249 छात्रछात्राओं का टीकाकरण पूर्ण किया गया तो सूर्या बालिका इंटर कालेज में 734, विद्यासागर इंटर कालेज 180, शहीद ध्रुवलाल मेमोरियल कालेज सुरियावां में 700, राजकीय हाईस्कूल अर्जुनपुर 73, तुलसी देवी बालिका उच्चतर माध्यमिक विद्यालय सुरियावां में 298, स्वामी विवेकानन्द उच्चतर माध्यमिक विद्यालय छनौरा 1085, विध्यवासिनी कान्वेंट स्कूल हरीपट्टी में सभी 60 का टीकाकरण पूर्ण कराया गया। उन्होंने अन्य विद्यालयों के शिक्षक, कर्मचारियों को टीकाकरण पूर्ण कराने पर जोर दिया। कहा कि संक्रमण से बचाव का यही सुरक्षित उपाय है। | hindi |
दूधिचुआ के मशीनी बेड़े में छ: नए डंपर अनपरा। संवाददाताएनसीएल दूधिचुआ क्षेत्र के मशीनी बेड़े में 6 नए डंपर शामिल हुए हैं । मंगलवार को एनसीएल के सीएमडी भोला सिंह, निदेशक वित्त एवं कार्मिक राम नारायण दुबे तथा निदेशक तकनीकी परियोजना एवं योजना एस एस सिन्हा ने दूधिचुआ ओसीपी ओपनकास्ट प्रोजेक्ट में आयोजित एक कार्यक्त्रम में 100 टन क्षमता के छ: डंपरों को झंडी दिखाकर राष्ट्र की उर्जा सुरक्षा के कार्य में नियोजित किया। प्रबन्धन के मुताबिक वर्ष 202324 में एनसीएल के 130 मिलियन टन लक्ष्य को ध्यान में रखते हुए हाल ही में 204 करोड़ रुपये में 100 टन क्षमता के 55 डंपरों का ऑर्डर दिया गया था जिनमें से 28 डंपरों को दूधिचुआ में तैनात किया जाएगा । यह छ: डंपर उन्हीं 28 डंपरों की खेप का ही हिस्सा हैं। दूधीचुआ क्षेत्र की उत्पादन क्षमता को बढ़ाकर 25 मिलियन टन किया जाना है, इसी लक्ष्य को हांसिल करने के लिए यहाँ पर 28 नए डंपरों की तैनाती की जा रही है। वर्तमान में दूधीचुआ क्षेत्र ने 22 मिलियन टन लक्ष्य के सापेक्ष अभी तक 17.26 मिलियन टन उत्पादन किया है।डंपर नवीनतम तकनीक व सुरक्षा सुविधाओं से लैस हैं और इनमें एर्गोनॉमिक तकनीक का इस्तेमाल किया गया है जिससे काम की गुणवत्ता और उत्पादन के साथ ही वहाँ पर तैनात चालक की सुविधा एवं सुरक्षा भी सुनिश्चित हो सकेगी और फलस्वरुप कार्य दशा में सुधार के साथ श्रम दक्षता भी बढ़ेगी । एनसीएल की 10 खुली खदानों में 1250 से अधिक भारी मशीनें तैनात हैं । एनसीएल वर्ष 202324 तक कोल इंडिया के 1 बिलियन टन कोयला उत्पादन में योगदान देने के लिए भारी मशीनों की खरीद पर लगभग 3000 करोड़ का निवेश कर रही है जिससे देश की ऊर्जा के क्षेत्र में आत्मनिर्भरता को बल मिलेगा For Hindustan : हिन्दुस्तान ईसमाचार पत्र के लिए क्लिक करें epaper.livehindustan.com | hindi |
*** اسۍ ما وٲتِتھ پێے پتھ ون *** پۯاران بہ اوسس راتس بلۍ | kashmiri |
പത്തനംതിട്ട ജില്ലയില് മഴ താണ്ഡവമാടുന്നു ശബരിഗിരി സംഭരണികളിലേക്ക് ശക്തമായ നീരൊഴുക്ക് PambaRiver പത്തനംതിട്ട മഹാപ്രളയത്തിനു സമാനമായി പത്തനംതിട്ടയില് മഴ തകര്ക്കുന്നു. തകര്ത്തു പെയ്ത മഴയില് ജില്ലയില് വ്യാപക നാശം. താഴ്ന്ന പ്രദേശങ്ങളില് വെള്ളം കയറി. പുനലൂര് മൂവാറ്റുപുഴ റോഡ്, റാന്നി ചെറുകോല്പ്പുഴ റോഡ്, മല്ലപ്പള്ളി ആനിക്കാട് റോഡ്, വകയാര്, മുറിഞ്ഞകല് എന്നിവിടങ്ങളില് ഗതാഗതം പൂര്ണമായും മുടങ്ങി. വ്യാപകമായി വീടുകള്ക്കും കൃഷി സ്ഥലങ്ങള്ക്കും നാശമുണ്ടായി. മണിമലയാറ്റില് ഒഴുക്കില് പ്പെട്ടയാളെ രക്ഷപ്പെടുത്തി. പന്തളം കുടശനാട് കാര് തോട്ടിലേക്കു മറിഞ്ഞെങ്കിലും വാഹനമോടിച്ചയാള് രക്ഷപ്പെട്ടു. മലയോര മേഖലകളിലേക്കുള്ള രാത്രി യാത്രയും വിനോദ സഞ്ചാരവും കലക്ടര് നിരോധിച്ചു. മണ്ണെടുപ്പും ക്വാറികളുടെ പ്രവര്ത്തനവും നിരോധിച്ചിട്ടുണ്ട്. ദുരന്ത നിവാരണ പ്രവര്ത്തനങ്ങളുടെ ഭാഗമായി ജില്ലയിലെ വില്ലേജ്, പഞ്ചായത്ത് ഓഫിസുകള് ഇന്നു തുറന്നു പ്രവര്ത്തിക്കാനും കലക്ടര് ദിവ്യാ എസ്.അയ്യര് ഉത്തരവിട്ടു. ഇന്നലെ രാവിലെ ജില്ലയില് റെഡ് അലര്ട്ട് പ്രഖ്യാപിച്ചിരുന്നു. ഇന്നു മഴയുടെ തീവ്രത കുറയുമെന്നാണ് പ്രതീക്ഷ. അതേസമയം, ജില്ലയില് ശക്തമായ മഴ ഇന്നും തുടരുമെന്നതിനാല് കാലാവസ്ഥ നിരീക്ഷണ കേന്ദ്രം യെലോ അലര്ട്ട് പ്രഖ്യാപിച്ചു. ആനത്തോട് ഡാമില് റെഡ് അലര്ട്ട് പ്രഖ്യാപിച്ചിട്ടുണ്ട്. ഡാം ഇന്നു തുറക്കാന് സാധ്യതയുള്ളതിനാല് പമ്ബാനദിയുടെ ഇരുകരകളിലും ജാഗ്രതാ നിര്ദേശം നല്കി. റാന്നിയുടെ താഴ്ന്ന പ്രദേശങ്ങള് വെള്ളത്തിനടിയിലാണ്. റാന്നി ചെത്തോങ്കര, എസ്സി സ്കൂള് ജംക്ഷന്, മാമുക്ക് പാലം, കെഎസ്ആര്ടിസി ബസ് സ്റ്റേഷന്, പ്രൈവറ്റ് ബസ് സ്റ്റാന്ഡ് എന്നിവിടങ്ങളില് വെള്ളം കയറി. വ്യാപാര സ്ഥാപനങ്ങളില് നിന്നു സാധനങ്ങള് സുരക്ഷിത സ്ഥാനത്തേക്കു മാറ്റാന് തുടങ്ങി. താഴ്ന്ന പ്രദേശങ്ങളിലെ വീടുകളില് കഴിയുന്നവര് ബന്ധുവീടുകളിലേക്കും മറ്റും മാറി തുടങ്ങി. 24 മണിക്കൂറില് 10 സെന്റീ മീറ്ററിലധികം മഴയാണ് ജില്ലയില് പെയ്തത്. റാന്നിയില് ജലനിരപ്പ് ഉയര്ന്ന സാഹചര്യത്തില് ചെറുകോല്പ്പുഴ, കോഴഞ്ചേരി, ആറന്മുള, തിരുവല്ല ഭാഗങ്ങളില് ജാഗ്രത നിര്ദേശം പുറപ്പെടുവിച്ചിട്ടുണ്ട്. അപ്പര് കുട്ടനാട്ടില് പ്രളയ സാധ്യത കണക്കിലെടുത്ത് മുന്കരുതല് നടപടികള് സ്വീകരിച്ചതായി കലക്ടര് അറിയിച്ചു. സീതത്തോട് ശബരിമല കാടുകളിലും ശബരിഗിരി ജല വൈദ്യുത പദ്ധതിയുടെ വൃഷ്ടി പ്രദേശങ്ങളിലും കനത്ത മഴ. ജല സംഭരണികളിലേക്കു ശക്തമായ നീരൊഴുക്ക്. ജല നിരപ്പ് 89 ശതമാനത്തില് എത്തി. മഴ തുടര്ന്നാല് ആനത്തോട് അണക്കെട്ടിന്റെ ഷട്ടറുകള് രാവിലെ തുറക്കാനുള്ള തയാറെടുപ്പില് വൈദ്യുതി ബോര്ഡ് അധികൃതര്. പമ്ബയില് 980.95 മീറ്ററും, കക്കിആനത്തോട് അണക്കെട്ടില് 978.72 മീറ്ററുമാണ് ജല നിരപ്പ്. 978.83ല് എത്തിയാല് ആനത്തോട് അണക്കെട്ടിന്റെ ഷട്ടറുകള് തുറക്കും. 6 മണിക്കൂറിനുള്ളില് ഒരടിയില് അധികം വെള്ളം ഉയര്ന്നു. ഏകദേശം 7 ദശലക്ഷം ഘനമീറ്റര് വെള്ളം ഒഴുകി എത്തി. പമ്ബയില് 12 മില്ലിമീറ്ററും കക്കിയില് 14 മില്ലിമീറ്ററും മഴ പെയ്തു. ആവശ്യമെങ്കില് ഷട്ടറുകള് തുറക്കുന്നതിനുള്ള തയാറെടുപ്പില് അണക്കെട്ട് സുരക്ഷാ വിഭാഗം ഉദ്യോഗസ്ഥര് സ്ഥലത്ത് ക്യാംപ് ചെയ്യുന്നുണ്ട്. ആനത്തോട് അണക്കെട്ട് തുറന്നാല് പമ്ബാ നദിയില് ജലനിരപ്പ് ഉയരും. ഇനിയും വെള്ളം സംഭരിക്കുന്നതിനുള്ള ശേഷി അണക്കെട്ടുകള്ക്കുണ്ട്. നിലവില് നദികളില് ഉയര്ന്ന ജലനിരപ്പ് തുടരുന്നതിനാല് തിടുക്കത്തില് ഷട്ടറുകള് തുറക്കുന്നതിനെപ്പറ്റി ചര്ച്ച നടക്കുന്നതേയുള്ളൂ. കക്കാട്ടാറ്റില് ജലനിരപ്പ് ക്രമാതീതമായി ഉയര്ന്നതിനെ തുടര്ന്ന് മൂഴിയാര്, മണിയാര് അണക്കെട്ടുകള് തുറന്നു. കക്കാട്ടാറ്റിലുള്ള അള്ളുങ്കല് ഇഡിസിഎല്, കാരിക്കയം അയ്യപ്പ ഹൈഡ്രോ ഇലക്ട്രിക് പെരുനാട് ജലവൈദ്യുത പദ്ധതികളുടെ ഷട്ടറുകള് ഉയര്ത്തിവച്ചിരിക്കുകയാണ്. കക്കാട് പദ്ധതിയുടെ വേലുത്തോട് തടയണ കവിഞ്ഞ് വെള്ളം ഒഴുകി. | malyali |
۱۹۸۸ پؠٹھٕ ۱۹۹۳ تام اوس سُہ پوُنے منٛز فِلم انسٹی ٹیوُٹُک ڈائریکٹر | kashmiri |
ಕೊರೋನಾಕ್ಕೆ ಬಲಿಯಾದವರ ದೇಹದಿಂದ ಅಂಗಾಂಗ ತೆಗೆಯಲಾಗುತ್ತಿದೆಯೆ?:ಲಕ್ಷ್ಮೀ ಹೆಬ್ಬಾಳ್ಕರ್ ಹುಬ್ಬಳ್ಳಿಆ.29: ಕೊರೋನಾ ಸೋಂಕು ದೃಢಪಟ್ಟ ಒಂದೆರಡು ದಿನಗಳಲ್ಲಿ ಕೆಲವರು ಮೃತಪಡುತ್ತಿದ್ದಾರೆ. ಅಂತ್ಯ ಸಂಸ್ಕಾರಕ್ಕೆ ಮನೆಯವರಿಗೆ ಅವಕಾಶ ನೀಡುತ್ತಿಲ್ಲ. ಹೀಗಾಗಿ ಮೃತಪಟ್ಟ ವ್ಯಕ್ತಿಯ ದೇಹದ ಯಾವುದಾದರೂ ಅಂಗಗಳನ್ನು ತೆಗೆಯಲಾಗುತ್ತಿದೆಯೆ? ಎಂಬ ಅನುಮಾನವನ್ನು ಸೋಂಕಿತರ ಕುಟುಂಬಸ್ಥರು ವ್ಯಕ್ತಪಡಿಸುತ್ತಿದ್ದಾರೆ ಎಂದು ಶಾಸಕಿ ಲಕ್ಷ್ಮೀ ಹೆಬ್ಬಾಳ್ಕರ್ ಹೇಳಿದ್ದಾರೆ. ನಗರದ ಭೈರಿದೇವರಕೊಪ್ಪದಲ್ಲಿ ಶುಕ್ರವಾರ ಆರೋಗ್ಯ ಅಭಯ ಹಸ್ತ ಕಾರ್ಯಕ್ರಮಕ್ಕೆ ಚಾಲನೆ ನೀಡಿ ಅವರು ಮಾತನಾಡಿದರು. ಹತ್ತು ಮಂದಿಗೆ ಸೋಂಕು ತಗುಲಿದರೆ ಅದರಲ್ಲಿ 5 ಜನ ಸಾಯುತ್ತಿದ್ದಾರೆ. ಅವರು ಕೊರೋನಾ ಕಾರಣಕ್ಕೆ ಸಾಯುತ್ತಿದ್ದಾರೋ ಅಥವಾ ಭಯಪಟ್ಟು ಮೃತಪಡುತ್ತಿದ್ದಾರೊ ಗೊತ್ತಿಲ್ಲ. ಕೊರೋನಾ ಹೆಸರಲ್ಲಿ ಹಣ ಲೂಟಿ ನಡೆಯುತ್ತಿರುವುದು ಎಲ್ಲರಿಗೂ ತಿಳಿದ ವಿಚಾರ. ಆಸ್ಪತ್ರೆಗಳು 300 ಚಿಕಿತ್ಸೆಗೆ 3 ರಿಂದ 5 ಲಕ್ಷ ಬಿಲ್ ಮಾಡುತ್ತಿದ್ದಾರೆ. ಕೋವಿಡ್ ದೃಢಪಟ್ಟ ಕೆಲವರು ಎರಡೇ ದಿನದೊಳಗೆ ಮೃತರಾಗುತ್ತಿದ್ದಾರೆ. ಅವರ ದೇಹವನ್ನು ಮನೆಯವರಿಗೂ ನೀಡದೆ ಅಂತ್ಯಸಂಸ್ಕಾರ ಮಾಡಲಾಗುತ್ತಿದೆ. ಹೀಗಾಗಿ ಕುಟುಂಬಸ್ಥರು ಅನುಮಾನ ವ್ಯಕ್ತಪಡಿಸುತ್ತಿದ್ದಾರೆ. ಇಲ್ಲಿ ಯಾರನ್ನು ದೂರಬೇಕು ತಿಳಿಯುತ್ತಿಲ್ಲ ಎಂದು ಆಕ್ರೋಶ ವ್ಯಕ್ತಪಡಿಸಿದರು. ಧಾರವಾಡ: ಖ್ಯಾತ ವೈದ್ಯ ಕರ್ಪೂರಮಠ ಇನ್ನಿಲ್ಲ ಜನರು ಕೊರೋನಾ ಕಾರಣಕ್ಕೆ ಸಾಕಷ್ಟು ಸಂಕಷ್ಟ ಎದುರಿಸುತ್ತಿದ್ದಾರೆ. ಆದರೆ, ರಾಜ್ಯ ಹಾಗೂ ಕೇಂದ್ರ ಸರ್ಕಾರಕ್ಕೆ ರಾಜಕೀಯವೇ ಮುಖ್ಯವಾಗಿದೆ. ಕಾಂಗ್ರೆಸ್ ಜನಸಾಮಾನ್ಯರ ಪರವಾಗಿ ನಿಲ್ಲುವ ಪಕ್ಷವಾಗಿದ್ದು, ನಮ್ಮ ಖಜಾನೆ ತುಂಬಿಸಿಕೊಳ್ಳಲು ಬಂದಿಲ್ಲ. ಮಾಜಿ ಪ್ರಧಾನಿ ವಾಜಪೇಯಿ ಒಬ್ಬ ಸಂಸದರಿಂದ ಸರ್ಕಾರ ಕಳೆದುಕೊಂಡಾಗ, ಖರೀದಿ ವ್ಯವಹಾರಕ್ಕೆ ಇಳಿಯದೆ ಆದರ್ಶ ಮೆರೆದರು. ಅವರ ಪ್ರಾಮಾಣಿಕ ರಾಜಕಾರಣ ಸ್ತುತ್ಯಾರ್ಹ ಎಂದರು. ಭಾರತದಲ್ಲಿ ಕೊರೋನಾ ವೈರಸ್ ಕಾಲಿಡುವುದಕ್ಕೆ ಮುನ್ನವೇ, ಫೆಬ್ರವರಿಯಲ್ಲೇ ರಾಹುಲ್ ಗಾಂಧಿ ಕೇಂದ್ರ ಸರ್ಕಾರಕ್ಕೆ ಎಚ್ಚರಿಕೆ ನೀಡಿದ್ದರು. ಕೂಡಲೇ ತುರ್ತು ಕ್ರಮ ಕೈಗೊಳ್ಳಲು ಸಲಹೆ ನೀಡಿದ್ದರು. ಆದರೆ, ಕೇಂದ್ರ ಸರ್ಕಾರ ಮಧ್ಯಪ್ರದೇಶದಲ್ಲಿ ಬಿಜೆಪಿ ಸರ್ಕಾರ ರಚಿಸುವ ಕಸರತ್ತು ನಡೆಸುತ್ತಿತ್ತು. ಏಕಾಏಕಿ ಲಾಕ್ಡೌನ್ ಘೋಷಣೆ ಮಾಡಿದ್ದರಿಂದ ಜನತೆ ಸಂಕಷ್ಟಅನುಭವಿಸಿದ್ದಾರೆ ಎಂದರು. ಜ್ವರ, ನೆಗಡಿ ಬಂದರೆ ಆಸ್ಪತ್ರೆಗೆ ತೆರಳಲು ಹಳ್ಳಿಯ ಜನ ಭಯ ಪಡುತ್ತಿದ್ದಾರೆ. ಅವರಿಗೆ ಕಾಂಗ್ರೆಸ್ ಆರೋಗ್ಯ ಅಭಯ ಹಸ್ತ ನೆರವಿಗೆ ಬರುತ್ತಿದೆ. ತರಬೇತಿ ಪಡೆದ ಕಾರ್ಯಕರ್ತರು ಜನರ ಆರೋಗ್ಯ ತಪಾಣೆ ಮಾಡುತ್ತಿದ್ದು, ಚಿಕಿತ್ಸೆಯ ಅಗತ್ಯವಿದ್ದರೆ ಪಕ್ಷದ ಆರೋಗ್ಯ ಸೆಲ್ ಮೂಲಕ ಆಸ್ಪತ್ರೆಯಲ್ಲಿ ಚಿಕಿತ್ಸೆ ನೀಡಲಾಗುತ್ತಿದೆ. ಹೀಗಾಗಿ ಯಾರೂ ಭಯ ಪಡುವ ಅಗತ್ಯವಿಲ್ಲ ಎಂದರು. ಈ ವೇಳೆ ಎಸ್ಸೆಸ್ಸೆಲ್ಸಿಯಲ್ಲಿ ಉತ್ತಮ ಅಂಕ ಪಡೆದ ವಿದ್ಯಾರ್ಥಿಯನ್ನು ಶಾಸಕಿ ಲಕ್ಷ್ಮೀ ಹೆಬ್ಬಾಳ್ಕರ್ ಸನ್ಮಾನಿಸಿದರು. ರಜತ್ ಉಳ್ಳಾಗಡ್ಡಿಮಠ, ಹನುಮಂತ, ಸಮೀರಖಾನ್, ವೀರಣ್ಣ ನೀರಲಗಿ ಇದ್ದರು. | kannad |
ஓ மணப்பெண்ணே விழாவில் கலந்து கொண்ட ப்ரியா பவானி சங்கர், ஹரிஷ் கல்யாண் புகைப்படங்கள்! ஓ மணப்பெண்ணே படவிழா சென்னையில் நடந்தது. இதில் படத்தின் நாயகி ப்ரியா பவானி சங்கர், நாயகன் ஹரிஷ் கல்யாண் உள்ளிட்டவர்கள் கலந்து கொண்டனர். ஓ மணப்பெண்ணே தெலுங்கில் வெளியாகி வெற்றி பெற்ற பெல்லி சூப்புலு படத்தின் தமிழ் ரீமேக். ரிது வர்மா, விஜய் தேவரகொண்டா நடித்த வேடத்தை தமிழில் ப்ரியா பவானி சங்கரும், ஹரிஷ் கல்யாண் செய்துள்ளனர். ஓ மணப்பெண்ணேயை கார்த்திக் சுந்தர் இயக்கியுள்ளார். படம் குறித்து பேசிய அவர், பெல்லி சூப்புலு படத்தை கெடுத்து விடாமல் தமிழுக்கு ஏற்றபடி மாற்றம் செய்துள்ளோம் என்றார். ஹரிஷ், ப்ரியா பவானி சங்கர் வந்த பிறகு படத்தின் பலம் கூடிவிட்டது. படத்துக்கு புதிதாக எதையும் நான் செய்யவில்லை என்றார் கார்த்திக் சுந்தர். நடிகை ப்ரியா பவானி சங்கர் பேசுகையில், இந்தப் படம் அனைவருக்கும் திருப்தி தந்த படம், ஹரிஷ் நல்ல ஒத்துழைப்பு தந்தார் என்றார். பெல்லி சூப்புலுவில் நாயகிக்கு அதிக முக்கியத்துவம் உண்டு. அதனை தனது பேச்சில் குறிப்பிட்ட ப்ரியா பவானி சங்கர், மற்ற படங்களில் ஹீரோவுக்கு ஜோடியாகத்தான் நாயகி பாத்திரம் இருக்கும். இந்தப் படத்தை தைரியமாக என்னுடைய படம் என்பேன் என்றார். ஓ மணப்பெண்ணே படத்தை திரையரங்குக்குப் பதில் டிஸ்னி ஹாட் ஸ்டார் ஓடிடியில் நேரடியாக வெளியிடுகின்றனர். விரைவில் வெளியீட்டு தேதி அறிவிக்கப்பட உள்ளது. | tamil |
Zomato के शेयर रिकार्ड हाई से 50 प्रतिशत नीचे, खरीदें या बेचें इंडिया न्यूज, नई दिल्ली: फूड डिलीवरी करने वाली कंपनी Zomato के शेयरों में गिरावट लगातार जारी है। आज 11 फरवरी को भी Zomato के शेयर में 6 प्रतिशत की गिरावट देखने को मिली है। बीते दिन यह शेयर 94.45 रुपए पर बंद हुआ था और आज यह 92.75 के भाव पर खुला। खुलते साथ बाजार में दबाव के चलते इसके शेयर में भी गिरावट आ गई। इंट्राडे में आज Zomato के शेयर ने 86.50 का निचला स्तर टच किया है। Zomato के शेयरों में गिरावट तिमाही नतीजों के बाद से देखने को मिल रही है। हालांकि इसकी लिस्टिंग शेयर बाजार में जबरदस्त हुई थी। लेकिन अब यह रिकॉर्ड हाई से लगभग 50 फीसदी गिर चुका है। कंपनी का घाटा हुआ कम Zomato Share Price जानकारी के मुताबिक Zomato को दिसंबर तिमाही में घाटा उठाना पड़ा है। हालांकि कंपनी अपना घाटा कम करने में कामयाब रही है। कंपनी को इस दौरान 63.2 करोड़ रुपये का घाटा हुआ जबकि एक साल पहले की समान तिमाही में कंपनी 352.6 करोड़ के नुकसान में थी। इसके अलावा कंपनी के रेवेन्यू में भी बढ़ोतरी हुई है और सालाना आधार पर यह 609.4 करोड़ से बढ़कर 1112 करोड़ रुपए हो गया है। बता दें कि Zomato का शेयर 66 प्रतिशत प्रीमियम पर लिस्ट हुआ था। इसकी लिस्टिंग 23 जुलाई 2021 को हुई थी। आईपीओ के लिए इश्यू प्राइस 76 रुपये था, जबकि यह 115 रुपये पर लिस्ट हुआ और 126 रुपये के भाव पर बंद हुआ था। अब तक यह शेयर 169 रुपये के रिकॉर्ड हाई पर पहुंचा है। इसके बाद से इस शेयर में गिरावट है। Also Read : गिरावट में खुली Share Market, सेंसेक्स में 900 अंकों की गिरावट Also Read : Monetary Policy Update आरबीआई ने ब्याज दरों में नहीं किया बदलाव, रेपो रेट 4 पर बरकरार Connect With Us : Twitter Facebook Youtube | hindi |
चीनी, वनस्पति नेपाल ले जाते पांच लोग पकडे़ गए चीनी, वनस्पति नेपाल ले जाते पांच लोग पकडे़ गए खनुआ। सशस्त्र सीमा बल 66वीं वाहिनी की सीमा चौकी खनुआ के जवानों ने गश्त के दौरान 13 बोरी चीनी, 5 पेटी वनस्पति, पांच बाइक के साथ पांच लोगों को पकड़ कर नौतनवां कस्टम को सौंप दिया। आरोपित चीनी एवं वनस्पति लेकर नेपाल जा रहे थे। खनुआ गांव के करीब जवानों ने पकड़ लिया। आरोपित कन्हैया यादव निवासी मदनपुर थाना मझगावा जिला रूपन्देही नेपाल, बृजेश ठाकुर निवासी बेथरी थाना बेथरी नेपाल, अभय यादव निवासी सेमरा थाना नौतनवां, जमुना यादव निवासी हाटीबनगाई नेपाल, सलीम अंसारी निवासी सियारी माई वार्ड नंबर 6 रूपन्देही नेपाल को पकड़ा गया। यह जानकारी कार्यवाहक कमांडेंट बरजीत सिंह ने दी है। | hindi |
నేను నీ దాన్ని.. నీవు నా వాడివి: సమంత పోస్ట్ వైరల్ నేను నీ దాన్ని.. నీవు నా వాడివి అంటూ సమంత తన ఇన్స్టాగ్రాం అకౌంట్లో షేర్ చేసిన ఓ పాత పోస్ట్ ఇప్పుడు వైరల్గా మారింది. సమంతనాగచైతన్య క్యూట్ కపుల్గా ఎంతో మంది అభిమానాన్ని సంపాదించుకున్నారు. అంతేకాదు సినీ వర్గాలు, అభిమానులు చూడముచ్చటైన జంట అని చెప్పుకున్నారు. టాలీవుడ్లో మోస్ట్ బ్యూటి్ఫుల్, రొమాంటిక్ కపుల్గా పేరు తెచ్చుకున్నారు చైసామ్. దాదాపు పదేళ్ల పరిచయం. ఏడేళ్ల ప్రేమలో ఆనందంగా గడిపి.. పెద్దలను ఒప్పించి..రెండు మత సాంప్రదాయాల ప్రకారం ఎంతో వైభవంగా పెళ్ళి చేసుకున్నారు. 2017 అక్టోబర్ 67 తేదీల్లో రెండు సంప్రదాయాల్లో సమంత నాగ చైతన్యల వివాహం జరిగిన సంగతి తెలిసిందే. ఇప్పుడు వారిద్దరు కలిసుంటే గనక నేడు అక్టోబర్7 నాలుగవ వివాహా వార్షికోత్సవం గ్రాండ్గా సెలబ్రేట్ చేసుకునేవారు. కానీ ఇటీవలే అక్టోబర్ 2 చైసామ్లు విడిపోతున్నట్లు ప్రకటించారు. అయితే, సమంత ఏడాది క్రితం పెట్టుకున్న ఓ ఇన్స్టా పోస్ట్ మరో సారి వైరల్గా మారింది. అన్నీ సవ్యంగా ఉండి ఉంటే ఈ క్యూట్ కపుల్ నేడు మ్యారేజ్ డే జరుపుకునేవారు. ఈ నేపథ్యంలో గతేడాది పెళ్లి రోజు సందర్భంగా, సమంత తన ఇన్స్టాగ్రామ్లో నాగ చైతన్యతో కలిసి ఉన్న ఫోటోను షేర్ చేసి.. నేను నీ దాన్ని.. నీవు నా వాడివి.. ఎలాంటి పరిస్థితులైనా రానీ.. మనిద్దరం కలసికట్టుగా వాటిని ఎదుర్కొందాం.. ఆహ్వానిద్దాం.. హ్యాపీ యానివర్సరీ హస్బెండ్ అని క్యాప్షన్తో పోస్ట్ చేశారు. సమంత షేర్ చేసిన ఈ పోస్ట్ ప్రస్తుతం బాగా వైరల్ అవుతోంది. | telegu |
Please send you applications to jurmalasbite [at] inbox [dot] lv naming team name, city and country representing and division. Registration deadline May 27th.
Minimum number of teams per division to be played is 5.
Let’s have an awesome opening for outdoor season! | english |
அடக்கடவுளே.! கள்ளத்தொடர்பில் இருந்த அக்கா.. துண்டு துண்டாக வெட்டிய தம்பி.. பிரபல நாட்டில் நடந்த பகீர் சம்பவம்..!! ஜெர்மனியில் திருமணம் முடிந்த பிறகு வேறொரு ஆணுடன் பழகிய தனது அக்காவை அவரது 2 தம்பிகளும் ஒன்றாக சேர்ந்து கொடூரமாக கொலை செய்த சம்பவம் அதிர்ச்சியை ஏற்படுத்தியுள்ளது. ஜெர்மனியில் 34 வயதுடைய ஆப்கன் நாட்டை சேர்ந்த இளம்பெண் ஒருவர் வசித்து வந்துள்ளார். இந்த இளம்பெண் தன்னுடைய கணவரை விட்டுவிட்டு மற்றொரு ஆணுடன் பழகி வந்துள்ளார். இதனை அறிந்த இளம்பெண்ணின் 2 தம்பிகளுக்கும் தனது அக்காவின் நடத்தை பிடிக்காமல் போயுள்ளது. ஆகையினால் அந்த இளம்பெண்ணை 2 தம்பிகளும் பெர்லினுக்கு வரவழைத்து அவருடைய கழுத்தை நெறித்தும், தொண்டையை துண்டாக வெட்டி கொடூரமாக கொலை செய்துள்ளார்கள். அதன்பின்பு மூத்த தம்பியின் வீட்டிற்கு அருகே கொடூரமாக கொன்ற தன்னுடைய அக்காவின் சடலத்தை புதைத்துள்ளார்கள். இந்நிலையில் இந்த கொடூர சம்பவம் தொடர்பாக காவல்துறை அதிகாரிகள் இளம்பெண்ணின் 2 தம்பிகளையும் கைது செய்து சிறையில் அடைந்துள்ளார்கள். | tamil |
ನರೇಗಾ ಕಾಮಗಾರಿ ಕೈಗೊಳ್ಳಲು ರೈತರಿಗೆ ಅವಕಾಶ ಯಳಂದೂರು: ಉದ್ಯೋಗ ಖಾತ್ರಿ ಯೋಜನೆಯಡಿ ರೈತರು ಭೂ ಅಭಿವೃದ್ಧಿ ಸೇರಿದಂತೆ ಇತರೆ ಕಾಮಗಾರಿ ಕೈಗೊಳ್ಳಲು ಅವಕಾಶವಿದ್ದು, ಕೃಷಿಕರು ಇದರ ಲಾಭ ಪಡೆದುಕೊಳ್ಳಬೇಕು ಎಂದು ಅಂಬಳೆ ಗ್ರಾಪಂ ಪಿಡಿಒ ಗಂಗಾಧರ್ ಮನವಿ ಮಾಡಿದರು. ಗ್ರಾಪಂ ಮುಂಭಾಗ ನಡೆದ ಸಾಮಾಜಿಕ ಲೆಕ್ಕ ಪರಿಶೋಧನೆ ಹಾಗೂ ಕ್ರಿಯಾಯೋಜನೆ ಅಭಿಯಾನದಲ್ಲಿ ಮಾತನಾಡಿದ ಅವರು, ರೈತರ ಕ್ರಿಯಾಯೋಜನೆ ಜಾರಿಯಲಿದ್ದು. ರೈತರಿಗೆ ಇರುವ ಮಾನದಂಡಗಳ ಬಗ್ಗೆ ಕಚೇರಿಯಲ್ಲಿ ಮಾಹಿತಿ ಲಭ್ಯವಿದೆ. ಈ ಕುರಿತು ಅರ್ಜಿ ಪಡೆದು, ಇದಕ್ಕೆ ನಿಗದಿಯಾಗಿರುವ ಪೆಟ್ಟಿಗೆಯಲ್ಲಿ ಹಾಕುವ ಮೂಲಕ ಕಾಮಗಾರಿಯನ್ನು ಪಡೆದುಕೊಳ್ಳಬಹುದು ಎಂದು ಸಲಹೆ ನೀಡಿದರು. ನರೇಗಾ ತಾಲೂಕು ಸಂಯೋಜಕ ರಾಜ್ಕುಮಾರ್ ಮಾತನಾಡಿ, ಪಂಚಾಯಿತಿ ವ್ಯಾಪ್ತಿಯ ಗ್ರಾಮಗಳಲ್ಲಿ 130 ಕಾಮಗಾರಿಗಳು ನಡೆದಿದ್ದು, ಒಂದುಕೋಟಿರೂ.ಗೂಅಧಿಕ ಹಣವನ್ನು ಪಾವತಿಸಲಾಗಿದೆ ಎಂದು ಮಾಹಿತಿ ನೀಡಿದರು. ರೇಷ್ಮೆ, ಪಶುಸಂಗೋಪನೆ, ಕೃಷಿ, ಮಹಿಳಾ ಮತ್ತು ಮಕ್ಕಳ ಅಭಿವೃದ್ಧಿ ಸೇರಿದಂತೆ ವಿವಿಧ ಇಲಾಖೆಗಳ ಅಧಿಕಾರಿಗಳು ತಮ್ಮ ಇಲಾಖೆಗಳ ಯೋಜನೆಗಳ ಬಗ್ಗೆ ಮಾಹಿತಿ ನೀಡಿದರು.ಅಂಬಳೆ ಗ್ರಾಮದಲ್ಲಿ ನಡೆದಿರುವ ಕೆಲ ನರೇಗಾ ಕಾಮಗಾರಿಗಳಲ್ಲಿ ಅವ್ಯವಹಾರ ನಡೆದಿದೆ. ಕೆರೆ ಹಾಗೂ ಕಾಲುವೆ ಕಾಮಗಾರಿಗೆ ಹೆಚ್ಚು ಹಣವನ್ನು ಪಾವತಿಸಲಾಗಿದೆ. ಕೆಲವರಿಗೆ ಇನ್ನೂ ನರೇಗಾದಡಿ ಕೂಲಿ ನೀಡಿಲ್ಲ ಎಂದು ಸ್ಥಳೀಯರು ದೂರು ಸಲ್ಲಿಸಿದರು. ಕಾಮಗಾರಿ ನಡೆಯದಿರುವ ಬಗ್ಗೆ ಲಿಖೀತ ದೂರು ಸಲ್ಲಿಸುವಂತೆ ಆಡಳಿತಾಧಿಕಾರಿ ಶ್ವೇತಾ ಸಲಹೆ ನೀಡಿದರು.ಸಭೆಯಲ್ಲಿ ತಾಪಂ ಸದಸ್ಯ ವೈ.ಕೆ.ಮೋಳೆ ನಾಗರಾಜು ಪಂಚಾಯಿತಿ ನೋಡಲ್ ಅಧಿಕಾರಿ ದೀಪಾ, ಪಿಡಿಒಗಂಗಾಧರ್, ಕಾರ್ಯದರ್ಶಿ ಪುಟ್ಟರಾಜು, ನಾಗರಾಜು, ಸತ್ಯಪ್ಪ. ಸರಸ್ವತಿ ಮತ್ತಿತರರು ಉಪಸ್ಥಿತರಿದ್ದರು. | kannad |
ലേബര് റൂമിലേക്ക് കയറിയപ്പോള് ഗൗരിയെ നഷ്ടപ്പെടുമോ എന്ന് ഞാന് ഭയപ്പെട്ടു: ജീവിതത്തിലെ ഓര്മിക്കാന് ഇഷ്ടപ്പെടാത്ത നിമിഷത്തെക്കുറിച്ച് ഷാരൂഖ് തന്റെ മൂത്തമകന് ആര്യന് ജനിക്കുന്ന സമയത്ത് ഏറെ പേടിച്ച് പോയ നിമിഷത്തെ കുറിച്ച് ബോളിവുഡിലെ കിംഗ് ഖാന് ഷാരുഖ് മനസ്സ് തുറക്കുകയാണ്.താരം പറഞ്ഞ കാര്യങ്ങളാണ് ഇപ്പോള് സോഷ്യല് മീഡിയയില് വൈറലായി മാറിക്കൊണ്ടിരിക്കുന്നത്. മകന് ജനിക്കുന്ന സമയത്ത് തങ്ങള് ഇരുവരും നിരവധി സ്വപ്നങ്ങള് നെയ്തു കൂട്ടിയിരുന്നു പക്ഷെ ഒന്പതാം മാസമായപ്പോഴേക്കും ,ഭാര്യ ഗൗരി യെ നഷ്ടപ്പെടുമോ എന്ന ഭയം തനിക്കേറെ ഉണ്ടായിരുന്നുവെന്നും ഷാരൂഖാന് പറയുന്നു. പ്രസവ വേദന അനുഭവിച്ചു കൊണ്ടിരിക്കുമ്ബോള് ഭാര്യക്ക് സിസേറിയന് വേണ്ടി വന്നിരുന്നു. അന്ന് ഭാര്യയുടെ കൂടെ ഓപ്പറേഷന് തിയേറ്ററില് കയറിയ അനുഭവവും താരം പങ്കുവെച്ചിരുന്നു. 1997 നവംബര് പതിമൂന്നിനാണ് മകന്ആര്യന് ഖാന് ജനിക്കുന്നത്. ആ സമയത്ത് തനിക്ക് എന്നെന്നേക്കുമായി ഭാര്യയെ നഷ്ടപ്പെട്ടേക്കും എന്ന് കരുതിയിരുന്നതായി ഷാരുഖ് പറയുന്നു. സിസേറിയനു കൊണ്ടു പോകുന്ന സമയത്ത് ശരീരമാസകലം ട്യൂബ് ഘടിപ്പിച്ചു , വളരെ പേടിപ്പെടുത്തുന്ന ഒരു അവസ്ഥയായിരുന്നു ഗൗരിക്ക് ഉണ്ടായിരുന്നത്. താന് ഓപ്പറേഷന് മുറിയില്നിന്ന് ശരിക്കും ഞെട്ടിപ്പോയിരുന്നു. ഗൗരിയെ തിരിച്ചുകിട്ടും എന്ന് ഒരു ഉറപ്പും ഇല്ലാത്ത അവസ്ഥയില് ആയിരുന്നു. മാതാപിതാക്കളെ നഷ്ടപ്പെട്ട തുകൊണ്ടുതന്നെ തനിക്ക് ആ വേദന നന്നായി അറിയാമായിരുന്നു. വളരെ ദുര്ബലമായിരുന്നു ഗൗരി.അവള് വേദന അനുഭവിക്കുന്നത് കണ്ടപ്പോള് സങ്കടം വന്നിരുന്നു.അപ്പോഴൊന്നും ജനിക്കാന് പോകുന്ന കുഞ്ഞിനെ കുറിച്ച് താന് ആലോചിച്ചില്ല. ഗൗരിയെ എങ്ങനെയെങ്കിലും തിരികെ കിട്ടണം എന്ന് മാത്രമേ ഞാന് വിചാരിച്ചിരുള്ളു എന്നും നടന് മനസ്സുതുറന്നു. | malyali |
ಎಬಿ ಡಿವಿಲಿಯರ್ಸ್ ದಂಪತಿಗೆ ಹೆಣ್ಣು ಮಗು, ಮೂರನೇ ಬಾರಿ ಅಪ್ಪನಾದ ಖುಷಿಯಲ್ಲಿ ಸ್ಫೋಟಕ ಬ್ಯಾಟ್ಸ್ಮನ್..! ಜೋಹಾನ್ಸ್ಬರ್ಗ್: ದಕ್ಷಿಣ ಆಫ್ರಿಕಾದ ಸ್ಫೋಟಕ ಬ್ಯಾಟ್ಸ್ಮನ್ ಎಬಿ ಡಿವಿಲಿಯರ್ಸ್ ಕುಟುಂಬ ಮತ್ತಷ್ಟು ವಿಸ್ತರಿಸಿಕೊಂಡಿದೆ. ಎಬಿಡಿ ಪತ್ನಿ ಡೇನೀಲ್ ನವೆಂಬರ್ 11 ರಂದು ಹೆಣ್ಣು ಮಗುವಿಗೆ ಜನ್ಮ ನೀಡಿದ್ದಾರೆ. ಈ ಕುರಿತು ಸ್ಫೋಟಕ ಬ್ಯಾಟ್ಸ್ಮನ್ ಎಬಿಡಿ ಇನ್ಸ್ಟಾಗ್ರಾಂನಲ್ಲಿ ಖುಷಿ ಹಂಚಿಕೊಂಡಿದ್ದಾರೆ. ಪತ್ನಿ ಹಾಗೂ ಮಗುವಿನೊಂದಿಗೆ ಇರುವ ಫೋಟೋವನ್ನು ಡಿವಿಲಿಯರ್ಸ್ ಪ್ರಕಟಿಸಿದ್ದಾರೆ. ಎಬಿಡಿ ದಂಪತಿಗೆ ಮೂರನೇ ಮಗು ಇದಾಗಿದ್ದು, ಇಬ್ಬರು ಗಂಡು ಮಕ್ಕಳಿದ್ದಾರೆ. ಹೆಣ್ಣು ಮಗುವಿಗೆ ಯೆಂಟೆ ಎಂದು ಹೆಸರಿಡಲಾಗಿದೆ. ಕ್ರಿಕೆಟ್ ವಲಯದಲ್ಲಿ ಎಬಿ ಡಿವಿಲಿಯರ್ಸ್ ಹಾಗೂ ಡೇನೀಲ್ ದಂಪತಿ ಸ್ಟಾರ್ ದಂಪತಿಗಳಾಗಿ ಗುರುತಿಸಿಕೊಂಡಿದ್ದಾರೆ. 2013ರಲ್ಲಿ ಮದುವೆಯಾದ ಈ ಜೋಡಿಗೆ 2015ರಲ್ಲಿ ಮೊದಲ ಮಗ ಅಬ್ರಹಾಂ ಡಿ ವಿಲಿಯರ್ಸ್ ಜನಿಸಿದರೆ, 2017ರಲ್ಲಿ ಜಾನ್ ಡಿ ವಿಲಿಯರ್ಸ್ ಜನಿಸಿದರು. ಇದೀಗ ಹೆಣ್ಣುಮಗುವಿನ ಜನನದಿಂದಾಗಿ ಎಬಿಡಿ ದಂಪತಿಯ ಸಂತೋಷ ಇಮ್ಮಡಿಗೊಂಡಿದೆ. ಇತ್ತೀಚೆಗೆ ಯುಎಇಯಲ್ಲಿ ಮುಕ್ತಾಯಗೊಂಡ ಐಪಿಎಲ್ನಲ್ಲಿ ಎಬಿಡಿ ಒಳಗೊಂಡ ಆರ್ಸಿಬಿ ಅಂಕಪಟ್ಟಿಯಲ್ಲಿ ನಾಲ್ಕನೇ ಸ್ಥಾನಕ್ಕೆ ತೃಪ್ತಿಪಟ್ಟುಕೊಂಡಿತು. 15 ಪಂದ್ಯಗಳನ್ನಾಡಿದ ಎಬಿಡಿ ಸ್ಟ್ರೈಕ್ರೇಟ್ 158.74 ರಂತೆ 5 ಅರ್ಧಶತಕ ಒಳಗೊಂಡಂತೆ 454 ರನ್ ಬಾರಿಸಿದ್ದರು. ಕುಟುಂಬ ಸದಸ್ಯರ ಜತೆ ಕಾಲ ಕಳೆಯುವ ಸಲುವಾಗಿ ಎಬಿ ಡಿವಿಲಿಯರ್ಸ್, ಮುಂಬರುವ ಬಿಗ್ ಬಾಷ್ ಲೀಗ್ನಿಂದ ಹೊರಗುಳಿದಿದ್ದಾರೆ. 2018ರಲ್ಲಿ ಅಂತಾರಾಷ್ಟ್ರೀಯ ಕ್ರಿಕೆಟ್ ಎಬಿಡಿ ವಿದಾಯ ಹೇಳಿದ್ದಾರೆ. 36 ವರ್ಷದ ಎಬಿ ಡಿವಿಲಿಯರ್ಸ್, ಟಿ20 ವಿಶ್ವಕಪ್ಗೆ ವಾಪಸ್ ವೇಳೆಗೆ ರಾಷ್ಟ್ರೀಯ ತಂಡಕ್ಕೆ ವಾಪಸಾಗುವ ನಿರ್ಧರಿಸಿದ್ದರೂ ಟೂರ್ನಿಯೇ ಮುಂದಿನ ವರ್ಷಕ್ಕೆ ಮುಂದೂಡಿಕೆಯಾಗಿದೆ. ಯುನಿವರ್ಸ್ ಬಾಸ್ ಖ್ಯಾತಿಯ ಕ್ರಿಸ್ ಗೇಲ್ಗೆ ಕಿಂಗ್ಸ್ ಇಲೆವೆನ್ ಪಂಜಾಬ್ ತಂಡದಲ್ಲಿ ಸಿಗಲಿದೆ ಹೆಚ್ಚಿನ ಅವಕಾಶ..! | kannad |
Welcome back. We hope you enjoyed the brief break from Excel Hell and got to cool off a bit in spreadsheet purgatory. Let’s continue with our investigation as to how the Financial Planning and Analysis organization inside your company got themselves into such hot “water used loosely”. Simply, FP&A relied on Excel to do what an EPM system has been doing for over two decades. Does this seem familiar? Maybe it’s time to catch-up. Catch up on part 1 here.
The manipulation and validation of data related to roll-ups within a Reporting Hierarchy is difficult. Rollups change during a planning process almost as a by-product given the nature of planning in the first place, i.e. to align the business operations the right way. Keeping up with the changes of these rolled up accounts, departments, projects, etc. often makes presenting the best numbers, at the expected time, impossible. EPM Tools like Host Analytics store hierarchical rollup relationships as segment hierarchies apart from the budgeted, forecasted, or actual amount data. Hierarchies are merely a separate reference point for the data so there are no rows to insert or delete or move around when a hierarchy changes within ANY segment. When a rollup changes, you manipulate the change in one place and every template (input) and report (output) are updated instantly.
There’s certain to be several consumers of budgeting and planning information. The various consumers along with various segments require several separate spreadsheets to produce all of the required reports. The FP&A organization is most likely opening and saving, updating a header or footer and saving and printing. Sometimes updates address the needs of 10-plus people throughout 20 different report types (sales, operations, manufacturing, R&D.) Within top analytics modules of EPM software are the capability to produce adhoc reports in minutes. Those reports can be pivoted, parsed, and summed on any hierarchical level within any dimension or segment needed for reporting. Further, there is the opportunity to place substitution variables in a report specification so that you can quickly execute the report for a slew of time periods, scenarios, or segment values. Finally, you can stack all of your report requests into a report collection in order to schedule and package your report distribution via a hands free process.
Getting the buy-in of individuals involved in the planning process is difficult because the mission is fractured and often modestly coordinated. Most participants are limited in scope and visibility and trust in the process and the underlying data is often suspect. Couple that with the fact that participants in the process find it time consuming and a veritable PITA. Once again, EPMs riding on a common architecture with the ability to offer the same construct to many participants simultaneously, establish a greater level of trust than spreadsheets. And from trust emerges buy-in not only to the process of collaboration to promote data input but also to the value of the data itself once it is produced.
Spreadsheets provide a perfect backdrop for performing analysis. Lots of what-if scenarios can emerge but the question always lies in the background as to whether or not the numbers in the spreadsheets are themselves the right ones. After all, some studies out of Harvard have reported that up to 88% of all spreadsheets contain errors. Just having an EPM won’t guarantee error-free computing but you will still have all of the what-if capabilities of spreadsheets. And with an EPM you only enter the data required for the overall model in one place. Therefore, there is a much greater chance that the numbers you play around with at the end are the best representation of the data you can get.
With spreadsheets FP&A folks sometimes struggle to find discrepancies resulting from incorrect input or formula calculations. The discrepancies could potentially be in an infinite number of places. EPM Systems, like most enterprise applications, contain audit trails, security, and data input logs that can be used to verify the data loaded into it. Whether the data was loaded manually or through an automated routine it’s in a self-contained system. There are a finite number of places for errors to hide whenever there are items that do not total or calculate exactly as expected.
In conclusion, Corporate Performance Measurement systems aka (EPM, CPM, FPM) have been in use for twenty plus years. I did come to find out that some organizations have done a bang-up job emulating a system with their usage of Excel. Inevitably, however, it was that desire to “systematize” excel that lead to the hell in the first place. Maybe it was the ‘cheaper alternative’ at the time. Perhaps it was ‘the time’ to implement. Most likely, it’s where ‘the current expertise’ laid. Fortunately all of that has been dealt with via subscription models, pay as you go, and three month EPM implementations, all just in time for the budget season to begin. If you want to see the first entry in this two-part series then click here: OMG, Excel Hell is Real, Part 1. | english |
અફઝલખાનની કબર તોડી નાંખવાની રાજ ઠાકરેની ધમકી, મહારાષ્ટ્ર સરકારે વધારી દીધી સુરક્ષા નવી દિલ્હી,તા 25 મે 2022,બુધવારમહારાષ્ટ્રમાં લાઉડ સ્પીકર વગાડવાના મુદ્દે ચાલી રહેલા ઘમાસાણ વચ્ચે મહારાષ્ટ્ર નવનિર્માણ સેનાના અધ્યક્ષ રાજ ઠાકરેએ ધમકી આપીને કહ્યુ છે કે, સતારમાં આવેલી અફઝલ ખાનની નાની કબર હવે મસ્જિદ બની ચુકી છે. જો રાજ્ય સરકાર તેને ધ્વસ્ત નહીં કરે તો અમારા કાર્યકરો તેને તોડી પાડશે.ઠાકરેએ કહ્યુ હતુ કે, એ વ્યક્તિ અમારા છત્રપતિ શિવાજી મહારાજની હત્યા કરવા માટે બીજાપુરથી આવ્યો હતો પણ ઉલટાનુ શિવાજી મહારાજે જ તેને મારી નાંખ્યો હતો. તેની કબર સતારામાં પ્રતાપગઢ કિલ્લા પાસે હતી. 6.5 ફૂટની કબર આજે 15000 ચોરસ ફૂટ વિસ્તારમાં ફેલાઈ ગઈ છે. તેના માટે કોણ જવાબદાર છે અને અહીંયા મસ્જિદ બની રહી છે તો તેને કોણ ફંડ આપી રહ્યુ છે?રાજ ઠાકરેની ધમકી બાદ મહારાષ્ટ્ર સરકારે કબરની સુરક્ષા વધારી દીધી છે. અહીંયા મોટા પાયે પોલીસ કર્મીઓને તૈનાત કરાયા છે.દરમિયાન સતારા પોલીસનુ કહેવુ છે કે, અફઝલ ખાનની કબર 2005થી જ પ્રતિબંધિત ક્ષેત્ર છે.આ પહેલા રાજ ઠાકરેની પાર્ટીએ ઔરંગઝેબની કબરને લઈને પણ ધમકી આપીને કહ્યુ હતુ કે, આ કબરને તોડી પાડવી જોઈએ જેથી ઔરંગઝેબના સંતાનો અહીંયા માથુ ટેકવા ના આવી શકે. | gujurati |
ಸಮಯಪ್ರಜ್ಞೆ ಮೆರೆದು ಮಾಲೀಕನ ಪ್ರಾಣ ಉಳಿಸಿತು ಗಿಳಿಮರಿ ಪ್ರಾಣಾಪಾಯದಲ್ಲಿದ್ದ ತನ್ನ ಮಾಲೀಕನನ್ನು ಗಿಳಿಯೊಂದು ರಕ್ಷಿಸಿದ ಘಟನೆ ಬ್ರಿಸ್ಬೇನ್ನಲ್ಲಿ ನಡೆದಿದೆ. ಆಂಟನ್ ನ್ಗುಗೇನ್ ಘಾಡ ನಿದ್ರೆಯಲ್ಲಿದ್ದರು. ಈ ವೇಳೆ ಅವರ ಮನೆಯಲ್ಲಿ ಆಕಸ್ಮಾತ್ ಆಗಿ ಬೆಂಕಿ ಕಾಣಿಸಿಕೊಂಡಿದೆ. ನ್ಗುಗೇನ್ ಮಾತ್ರ ಇದರ ಪರಿವೇ ಇಲ್ಲದೆ ನಿದ್ರಿಸುತ್ತಿದ್ದರು. ಆದ್ರೆ ಅವರ ಮನೆಯಲಿದ್ದ ಗಿಳಿ ಎರಿಕ್ ಕೂಡಲೇ ಸದ್ದು ಗದ್ದಲ ಮಾಡಿ ತನ್ನ ಮಾಲೀಕನನ್ನು ಎಚ್ಚರಿಸಿದೆ. ಎರಿಕ್ ಕಿರುಚಾಡುವುದನ್ನು ಕೇಳಿ ನಿದ್ದೆಯಿಂದ ಎದ್ದ ನ್ಗುಗೇನ್ ಮನೆಯ ಒಂದು ಬದಿಯಲ್ಲಿ ಬೆಂಕಿ ಉರಿಯುತ್ತಿರುವುದನ್ನು ಕಂಡು ಶಾಕ್ ಆಗಿದ್ದಾರೆ. ಕೂಡಲೇ ಅವರು ಎರಿಕ್ನನ್ನು ಎತ್ತಿಕೊಂಡು ಮನೆಯಿಂದ ಆಚೆ ಓಡಿದ್ದಾರೆ. ಕೂಡಲೇ ಸ್ಥಳೀಯ ಪೊಲೀಸರಿಗೆ ಮಾಹಿತಿ ನೀಡಿ ಬೆಂಕಿ ಆರಿಸುವಲ್ಲಿ ಯಶಸ್ವಿಯಾಗಿದ್ದಾರೆ. ಗಿಳಿಯ ಸಮಯ ಪ್ರಜ್ಞೆಯಿಂದ ಮಾಲೀಕನ ಜೀವ ಉಳಿದಿದೆ ಎಂದು ಜನರು ಮೆಚ್ಚುಗೆ ವ್ಯಕ್ತಪಡಿಸಿದ್ದಾರೆ. | kannad |
Comfortable Flight Journey: ৫ টোটকা: বিমানযাত্রা হয়ে উঠবে বিলাসবহুল কর্মব্যস্ত জীবনের ক্লন্তি দূর করতে অফিস থেকে দিন সাতেকের ছুটি নিয়েছেন? বেড়াতে যাওয়ার সব প্রস্তুতি ইতিমধ্যেই সেরে ফেলেছেন? আগে থেকেই ভেবে রেখেছেন এই ছুটিতে বিলাসিতার সঙ্গে কোনও রকম আপস করবেন না! তবে বিলাসবহুল হোটেল বুক করতে গিয়ে খরচ অনেকটাই বেশি হয়ে গিয়েছে তাই ইচ্ছা থাকলেও বিমানের বিজনেস ক্লাসের টিকিট আর কাটা হল না! চিন্তা নেই কয়েকটি ফন্দিফিকির জানলেই ইকনমি ক্লাসে যাত্রা করেও ভোগ করতে পারেন বিজনেস ক্লাসের অনুভূতি ভাবছেন এ কী করে সম্ভব? ১ বিমানের এক্সিট রোতে আসন বুক করুন: এ ক্ষেত্রে আপনাকে আগে থেকেই বুকিং সারতে হবে এর জন্য দিতে হতে পারে কিছুটা বাড়তি টাকাও তা ছাড়া চেক ইন করার সময়েও নিজের পছন্দের আসনটি বেছে নিতে পারেন বিমানে যাত্রার সময়ে পা বেশি ছড়ানো যায় না বলে অনকেরই অভিযোগ থাকে বিমানের গেটে ওঠার মুখেই যে আসনগুলি থাকে, সেই সিটগুলির সামনে অনেকটা বাড়তি জায়গা থাকে তাই পা ছড়িয়ে বসতে কোনও সমস্যা হয় না ২ স্লিপিং মাস্ক নিতে ভুলবেন না: বিমান যাত্রার সময়ে অনেকেই স্লিপিং মাস্ক নিতে ভুলে যান দীর্ঘ বিমানযাত্রার ক্ষেত্রে এই ভুল করলে আপশোসের শেষ থাকে না বিমানে অনেকের ঘুম আসতে চায় না সে ক্ষেত্রে একটি স্লিপিং মাস্ক ব্যবহার করে দেখতেই পারেন ৩ বিমানে ওঠার আগেই পছন্দের খাবারটি কিনে নিন: বিমানের খাবারের স্বাদ বেশির ভাগ সময়েই ভাল থাকে না বিজনেস ক্লাসে যাত্রা করলে যে মানের খাবার পাওয়া য়ায় ইকনমি ক্লাসের ক্ষেত্রে তা নয় তা ছাড়া, বিমানে যে কোনও খাবারের দাম থাকে আকাশ ছোঁয়া তাই বিমানে ওঠার আগে নিজের পছন্দের খাবারটি কিনে নেওয়াই শ্রেয় তবে আপনি যে বিমানে যাত্রা করবেন, তাতে খাবার নিয়ে ওঠা যাবে কি না তা অবশ্যই জেনে নেবেন ৪ পোশাকের স্বচ্ছন্দের সঙ্গে আপোস নয়: দীর্ঘ ক্ষণের বিমানযাত্রার সময়ে ঢিলেঢালা পাজামা আর টিশার্ট পরাই ভাল বিমান যাত্রার সময়ে জিন্স না পরাই ভাল ঢিলেঢালা পোশাকে মিলবে আরাম ৫ নেক পিলো অবশ্যই সঙ্গে রাখুন: বিমানযাত্রার সময়ে সঙ্গে নেক পিলো রাখতে ভুলবেন না যেন নইলে ঘুমনোর সময়ে ঘাড়ে ব্যথা হতে পারে এটি সঙ্গে রাখলে কিন্তু আপনার ট্রলিটির বোঝা মোটেই বাড়বে না হ্যান্ডব্যাগের সঙ্গেই এটি ক্লিপ করে নিতে পারেন | bengali |
ज्वाइंट मजिस्ट्रेट ने शराब की दुकानों का किया निरीक्षण जागरण संवाददाता, नौपेड़वा जौनपुर : ज्वाइंट मजिस्ट्रेट उपजिलाधिकारी सदर हिमांशु नागपाल व सीओ सदर रणविजय सिंह ने रविवार को शराब की दुकानों का निरीक्षण किया। इस दौरान अगलबगल के दुकानदारों में खलबली मच गई। शाम लगभग पांच बजे ज्वाइंट मजिस्ट्रेट नौपेड़वा बाजार स्थित शराब की दुकान पर पहुंचे तो अफरातफरी मच गई। अधिकारियों की तरफ से बिक्री के लिए रखी शराब की बारीकी से जांच की। बिक्री रेट सूची न रहने पर नाराजगी जताई। मजिस्ट्रेट ने दुकानदारों से बिक्री रेट सूची लगाने का निर्देश दिया। अधिकारी मई गांव स्थित शराब की दुकान पर पहुंचे जहां बाहर शराब की खाली बोतलों को देख नाराजगी जताई। इस मौके पर थानाध्यक्ष बक्शा दिव्य प्रकाश सिंह आदि मौजूद थे। | hindi |
These huge prehistoric fish are one of the most rewarding challenges in the river.
The sturgeon is a term for a genus of fish (Acipenser) of which twenty species are known. One of the oldest fish in existence, which they are generally known as dinosaurs fishing. Sturgeons ranging from 8 a 11 pies (2,5 a 3,5 m) long they are not uncommon, by any means and some species grow to a much larger size.
With its snout remove the soft bottom, and their sensitive barbels detect shells, crustaceans and small fish of feeding. Having no teeth, They are not able to take larger prey.
sturgeons, mariones o sollos (Acipenser spp.) Acipenseridae family belong to the genus of ray-finned fish of about twenty species, ventral mouth, five longitudinal rows of plates, tracing rivers to spawn, You can reach 3.5 m length and 350 kilos weight, and whose eggs caviar is prepared.
Some of the best known species of sturgeon are common sturgeon (Acipenser sturio), whitefish (Acipenser transmontanus), the lake sturgeon (Acipenser fulvescens) and beluga sturgeon (huso huso).
White sturgeon is one of the largest and sport fishing It is the most extreme.
Bait for sturgeon fishing will be determined by the time of year. Sturgeon seek specific foods according to the season.
Early spring, lamprey eels, the trench and eel are good olachen. Later, in the spring, the olachen is the main bait. Early summer lamprey and eel, eels trench with squawfish you can fish thrown in August .During, Sockeye fishing is based, Sockeye parts and roes . Autumn is a time of plenty for sturgeon and bag food is varied. Pink salmon roe, and Chum white sturgeon roe roe spring are excellent baits that worked until well into December.
In the winter months sturgeons are very inactive due to cold water temperatures and fishing is spotty with any bait.
In Spain there is only the common sturgeon that lives in coastal marine waters, but from late winter to spring it ascends rivers to spawn. Because of environmental changes, It is considered an endangered species.
Hello, I wanted to ask a question… Do these fish can be located on the rivers of Catalonia “ebro” O “Tue”? Please, that inhabit rivers? ” as close to Catalonia and Aragon, Thank you! | english |
गडचिरोली जिले में ऑनलाईन पढ़ाई की क्या शिकायतें हैं? नागपुरदि.19 गडचिरोली जिले के दुर्गम भागों में ऑनलाईन शिक्षण व मध्यान्ह भोजन के संदर्भ में विद्यार्थियों की क्या शिकायतें है, इसकी जानकारी लेकर एक सप्ताह में रिपोर्ट प्रस्तुत करने के निर्देश मुंबई उच्च न्यायालय के नागपुर खंडपीठ ने राज्य सरकार को शुक्रवार को दिए. मामले पर न्यायमूर्तिद्वय नितीन जामदार व अनिल पानसरे के समक्ष सुनवाई हुई. 2020 में गडचिरोली जिले के 10 शालेय विद्यार्थियों ने उच्च न्यायालय को पत्र लिखकर ऑनलाइन पढ़ाई व मध्यान्ह भोजन उपलब्ध नहीं होने की शिकायत की थी. जिसके चलते इस संदर्भ में न्यायालय ने स्वयं ही जनहित याचिका दाखल की है. केंद्रीय दूरसंचार विभाग ने न्यायालय में दाखल किए प्रतिज्ञा पत्र के अनुसार गडचिरोली जिले के 829 से अधिक गांवों में इंटरनेट सुविधा नहीं तो अॅन्युअल स्टेटस ऑफ एजुकेशन रिपोर्ट असर2020 अनुसार ग्रामीण भागों में सिर्फ 76.3 प्रतिशत विद्यार्थियों के पास ही स्मार्ट फोन है. वहीं अनेक गांवों में 24 घंटे बिजली उपलब्ध नहीं रहती. ऑनलाईन पढ़ाई के लिए स्मार्ट फोन, इंटरनेट व बिजली यह तीन सुविधा आवश्यक है. लेकिन ये सुविधाएं न होने से हजारों विद्यार्थी ऑनलाइन पढ़ाई से वंचित है.बावजूद इसके विद्यार्थी फिलहाल स्कूल में नहीं जाने से उन तक मध्यान्ह भोजन या इसके बदले पैसे पहुंचाने के लिए उपाय योजना भी नहीं की गई. एड. फिरदौस मिर्जा ने न्यायालय मित्र के रुप में काम देखा. | hindi |
<?php
namespace MediaAlchemyst\Tests\Transmuter;
use MediaAlchemyst\Transmuter\Document2Image;
use MediaAlchemyst\Tests\AbstractAlchemystTester;
use MediaAlchemyst\DriversContainer;
use MediaAlchemyst\Specification\Image;
use MediaAlchemyst\Specification\Video;
use Symfony\Component\Process\ExecutableFinder;
class Document2ImageTest extends AbstractAlchemystTester
{
/**
* @var Document2Image
*/
protected $object;
protected $specs;
protected $source;
protected $dest;
protected function setUp()
{
$executableFinder = new ExecutableFinder();
if (!$executableFinder->find('unoconv')) {
$this->markTestSkipped('Unoconv is not installed');
}
$this->object = new Document2Image(new DriversContainer(), $this->getFsManager());
$this->specs = new Image();
$this->source = $this->getMediaVorus()->guess(__DIR__ . '/../../../files/Hello.odt');
$this->dest = __DIR__ . '/../../../files/output.jpg';
}
/**
* Tears down the fixture, for example, closes a network connection.
* This method is called after a test is executed.
*/
protected function tearDown()
{
if (file_exists($this->dest) && is_writable($this->dest)) {
unlink($this->dest);
}
}
/**
* @covers MediaAlchemyst\Transmuter\Document2Image::execute
*/
public function testExecute()
{
$this->specs->setDimensions(320, 240);
$this->specs->setResizeMode(Image::RESIZE_MODE_INBOUND);
$this->object->execute($this->specs, $this->source, $this->dest);
$MediaDest = $this->getMediaVorus()->guess($this->dest);
$this->assertEquals(320, $MediaDest->getWidth());
$this->assertEquals(240, $MediaDest->getHeight());
}
/**
* @covers MediaAlchemyst\Transmuter\Document2Image::execute
* @covers MediaAlchemyst\Exception\SpecNotSupportedException
* @expectedException MediaAlchemyst\Exception\SpecNotSupportedException
*/
public function testExecuteWrongSpecs()
{
$this->specs = new Video();
$this->object->execute($this->specs, $this->source, $this->dest);
}
}
| code |
Awk, Replace character if not followed by?
Replace character if not followed by?
> Replace all ac by a char-combination which is not used otherwise.
Thanks. Just can't seem to get my head into gear today!!
2. string.replace() can't replace newline characters???
3. REPLACE not replacing under Win 95?? | english |
/*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package com.facebook.presto.server;
import com.facebook.presto.Session;
import com.facebook.presto.SessionRepresentation;
import com.facebook.presto.execution.ExecutionFailureInfo;
import com.facebook.presto.execution.QueryInfo;
import com.facebook.presto.execution.QueryState;
import com.facebook.presto.spi.ErrorCode;
import com.facebook.presto.spi.ErrorType;
import com.facebook.presto.spi.PrestoWarning;
import com.facebook.presto.spi.QueryId;
import com.facebook.presto.spi.memory.MemoryPoolId;
import com.facebook.presto.spi.resourceGroups.QueryType;
import com.facebook.presto.spi.resourceGroups.ResourceGroupId;
import com.fasterxml.jackson.annotation.JsonCreator;
import com.fasterxml.jackson.annotation.JsonProperty;
import com.google.common.collect.ImmutableList;
import javax.annotation.Nullable;
import javax.annotation.concurrent.Immutable;
import java.net.URI;
import java.util.List;
import java.util.Optional;
import static com.facebook.presto.execution.QueryState.FAILED;
import static com.facebook.presto.memory.LocalMemoryManager.GENERAL_POOL;
import static com.facebook.presto.server.BasicQueryStats.immediateFailureQueryStats;
import static com.google.common.base.MoreObjects.toStringHelper;
import static java.util.Objects.requireNonNull;
/**
* Lightweight version of QueryInfo. Parts of the web UI depend on the fields
* being named consistently across these classes.
*/
@Immutable
public class BasicQueryInfo
{
private final QueryId queryId;
private final SessionRepresentation session;
private final Optional<ResourceGroupId> resourceGroupId;
private final QueryState state;
private final MemoryPoolId memoryPool;
private final boolean scheduled;
private final URI self;
private final String query;
private final BasicQueryStats queryStats;
private final ErrorType errorType;
private final ErrorCode errorCode;
private final ExecutionFailureInfo failureInfo;
private final Optional<QueryType> queryType;
private final List<PrestoWarning> warnings;
@JsonCreator
public BasicQueryInfo(
@JsonProperty("queryId") QueryId queryId,
@JsonProperty("session") SessionRepresentation session,
@JsonProperty("resourceGroupId") Optional<ResourceGroupId> resourceGroupId,
@JsonProperty("state") QueryState state,
@JsonProperty("memoryPool") MemoryPoolId memoryPool,
@JsonProperty("scheduled") boolean scheduled,
@JsonProperty("self") URI self,
@JsonProperty("query") String query,
@JsonProperty("queryStats") BasicQueryStats queryStats,
@JsonProperty("errorType") ErrorType errorType,
@JsonProperty("errorCode") ErrorCode errorCode,
@JsonProperty("failureInfo") ExecutionFailureInfo failureInfo,
@JsonProperty("queryType") Optional<QueryType> queryType,
@JsonProperty("warnings") List<PrestoWarning> warnings)
{
this.queryId = requireNonNull(queryId, "queryId is null");
this.session = requireNonNull(session, "session is null");
this.resourceGroupId = requireNonNull(resourceGroupId, "resourceGroupId is null");
this.state = requireNonNull(state, "state is null");
this.memoryPool = memoryPool;
this.errorType = errorType;
this.errorCode = errorCode;
this.failureInfo = failureInfo;
this.scheduled = scheduled;
this.self = requireNonNull(self, "self is null");
this.query = requireNonNull(query, "query is null");
this.queryStats = requireNonNull(queryStats, "queryStats is null");
this.queryType = requireNonNull(queryType, "queryType is null");
this.warnings = requireNonNull(warnings, "warnings is null");
}
public BasicQueryInfo(
QueryId queryId,
SessionRepresentation session,
Optional<ResourceGroupId> resourceGroupId,
QueryState state,
MemoryPoolId memoryPool,
boolean scheduled,
URI self,
String query,
BasicQueryStats queryStats,
ExecutionFailureInfo failureInfo,
Optional<QueryType> queryType,
List<PrestoWarning> warnings)
{
this(
queryId,
session,
resourceGroupId,
state,
memoryPool,
scheduled,
self,
query,
queryStats,
(failureInfo != null && failureInfo.getErrorCode() != null) ? failureInfo.getErrorCode().getType() : null,
failureInfo != null ? failureInfo.getErrorCode() : null,
failureInfo,
queryType, warnings);
}
public BasicQueryInfo(QueryInfo queryInfo)
{
this(queryInfo.getQueryId(),
queryInfo.getSession(),
queryInfo.getResourceGroupId(),
queryInfo.getState(),
queryInfo.getMemoryPool(),
queryInfo.isScheduled(),
queryInfo.getSelf(),
queryInfo.getQuery(),
new BasicQueryStats(queryInfo.getQueryStats()),
queryInfo.getErrorType(),
queryInfo.getErrorCode(),
queryInfo.getFailureInfo(),
queryInfo.getQueryType(),
queryInfo.getWarnings());
}
public static BasicQueryInfo immediateFailureQueryInfo(Session session, String query, URI self, Optional<ResourceGroupId> resourceGroupId, ExecutionFailureInfo failure)
{
return new BasicQueryInfo(
session.getQueryId(),
session.toSessionRepresentation(),
resourceGroupId,
FAILED,
GENERAL_POOL,
false,
self,
query,
immediateFailureQueryStats(),
failure,
Optional.empty(),
ImmutableList.of());
}
@JsonProperty
public QueryId getQueryId()
{
return queryId;
}
@JsonProperty
public SessionRepresentation getSession()
{
return session;
}
@JsonProperty
public Optional<ResourceGroupId> getResourceGroupId()
{
return resourceGroupId;
}
@JsonProperty
public QueryState getState()
{
return state;
}
@JsonProperty
public MemoryPoolId getMemoryPool()
{
return memoryPool;
}
@JsonProperty
public boolean isScheduled()
{
return scheduled;
}
@JsonProperty
public URI getSelf()
{
return self;
}
@JsonProperty
public String getQuery()
{
return query;
}
@JsonProperty
public BasicQueryStats getQueryStats()
{
return queryStats;
}
@Nullable
@JsonProperty
public ErrorType getErrorType()
{
return errorType;
}
@Nullable
@JsonProperty
public ErrorCode getErrorCode()
{
return errorCode;
}
@Nullable
@JsonProperty
public ExecutionFailureInfo getFailureInfo()
{
return failureInfo;
}
@JsonProperty
public Optional<QueryType> getQueryType()
{
return queryType;
}
@JsonProperty
public List<PrestoWarning> getWarnings()
{
return warnings;
}
@Override
public String toString()
{
return toStringHelper(this)
.add("queryId", queryId)
.add("state", state)
.toString();
}
}
| code |
હવે ગરમીઓમાં બિંદાસ વાપરો AC! આ ડિવાઇસને ખરીદીને ઘટાડો લાઇટબિલ How to Reduce Electricity Bill in Summers: ગરમી દરરોજ વધતી જાય છે અને ઘરમાં એક કલાક પણ એસી વિના રહી શકાતું નથી. પરંતુ દિવસભર એસી ચલાવવાનો અર્થ છે કે મહિનાના અંતે મોંઘું લાઇટ બિલ ભરવાની તૈયારી. જો તમે પણ આ સમસ્યાનો સામનો કરી રહ્યા છો તો અમારી પાસે આ સમસ્યા માટે એક કમાલનું સમાધાન છે. આજે અમે તમને એક એવા ડિવાઇસ વિશે જણાવીશું જેનાથી તમે દિવસભર એસી ચલાવ્યા બાદ પણ તમે વિજળ બિલને ઓછું કરી શકો છો. આવો જાણીએ અમે કયા ડિવાઇસની વાત કરી રહ્યા છીએ... ઘરે લાવો આ ડિવાઇસ અને વિજળીના બિલમાં કરો બચત તમને જણાવી દઇએ કે અમે અહીં એસી પાવર સેવર ડિવાઇસની વાત કરી રહ્યા છીએ, જેનો ઉપયોગ કરીને તમે તમારા વિજબિલને આરામથી ઓછું કરી શકો છો અને તમારે અટકી અટકીને એસી ચલાવવું નહી પડે. આ એક એવું ડિવાઇસ છે જેને તમે તમારા એસીમાં ફિટ કરી શકો છો અને વિજળી બચાવવામાં મદદ કરશે. Photo Credit: Amazon, IndiaMart 26 દિવસ સુધી આ 5 રાશિવાળા પર થશે ધનવર્ષા, જીવનમાં વધશે રોમાન્સ! તમે પણ છો સામેલ? સૌથી પહેલાં વાત કરીએ Dynamic AC Power Saver Device ની, તે એક સારો ઓપ્શન છે. આ એક ફૂલ્લી ઓટોમેટિક એસી પાવર સેવર ડિવાઇસ છે, જેને તમે કોઇ એન્જીનિયરની મદદથી તમારા એસીમાં લગાવી શકો છો. તમને જણાવી દઇએ કે આ ડિવાઇસને લગાવીને તમે તમારા ઘરના લાઇટ બિલ પર 10 થી 40 નો ઘટાડો કરી શકો છો. બીજા ડિવાઇસના મુકાબલે ઘણું મોંઘું છે અન તેને 7800 રૂપિયામાં ખરીદી શકાય છે. 799 રૂપિયામાં AC Power Saver Device Proelectra MDP08 એક સારો ઓપ્શન છે, જેને તમે વિજળી બચાવવા માટે ઉપયોગ કરી શકો છો. 1KW ની કેપેસિટીવાળા આ વિજળી બચાવનાર ડિવાઇસને ઘરમાં અથવા ઓફિસ, ક્યાંય પણ ઉપયોગ કરી શકો છો. અમેઝોન પર તેની કિંમત 2200 રૂપિયા છે પરંતુ 64 ટકા ડિસ્કાઉન્ટ બાદ તેને 799 રૂપિયામાં ખરીદી શકો છો. તેને તમે બસ તમારા એસીમાં ફિટ કરાવવું પડશે અને કામ થઇ જશે. , : twitter | gujurati |
c$glsph
c------------------------------------------------------------
double precision function glsph(cl , m , id)
c------------------------------------------------------------
c
c FUNCTION SUBROUTINE TO COMPUTE GENERALIZED SPHERICAL HARMONICS
c FUNCTION BELOW 4TH DEGREE
c
implicit double precision (a-h,o-z)
c
c1 = dcos(cl)
c2 = c1*c1
c4 = c2*c2
c
if (m.eq.0) then
c
if (id.eq.1) then
a1 = 3.d0/(4.d0*dsqrt(5.d0))
a2 = 35.d0*c4 - 30.d0*c2 + 3.d0
a3 = 1.d0 - 3.d0*c2 !!
glsph = a1*a2/a3
else if (id.eq.2) then
a1 = 2.d0/dsqrt(5.d0)
a2 = 1.d0
a3 = 1.d0 - 3.d0*c2 !!
glsph = a1*a2/a3
endif
c
else if (m.eq.1) then
c
a1 = dsqrt(3.d0)/(2.d0*dsqrt(2.d0))
a2 = 7.d0*c2 - 3.d0
glsph = a1*a2
c
else if (m.eq.2) then
c
a1 = dsqrt(3.d0)/2.d0
a2 = 7.d0*c2 - 1.d0
glsph = a1*a2
c
endif
c
return
end
c
| code |
Good News: लोकतंत्र का नया चेहरा... पंचायत चुनाव से पहले ग्रामीणों ने उम्मीदवारों की ली लिखित परीक्षा रांची, डिजिटल डेस्क। झारखंड से सटे ओडिशा में पंचायत चुनाव की प्रक्रिया शुरू हो चुकी है। इस दौरान सुंदरगढ़ जिले की मालूपाड़ा गांव में लोकतंत्र का नया चेहरा देखने को मिल रहा है। यहां चुनाव से पहले ही गांव वालों ने सरपंच पद का चुनाव लड़ रहे उम्मीदवारों की लिखित और मौखिक परीक्षा ली है। परीक्षा का मकसद भी साफ है। ग्रामीण जानना चाहते हैैं कि कौन उम्मीदवार उनकी समस्याओं को दूर करने के प्रति कितना गंभीर है। चुनाव के पहले वादों की झड़ी लगा देने वाले ज्यादातर उम्मीदवार इस परीक्षा में ग्रामीणों के सवालों का जवाब देने में हांफते नजर आए। परीक्षा में शामिल आठ प्रत्याशियों में पांच फेल हो गए, जबकि तीन को ग्रामीणों ने उत्तीर्ण घोषित किया है। ग्रामीणों द्वारा प्रत्याशी चयन का यह अनोखा तरीका चर्चा का विषय बना हुआ है। इस पंचायत में 18 फरवरी को चुनाव होना है।आदिवासी बहुल गांव में लोग समझ रहे वोट की अहमियत कुतरा पंचायत का यह गांव आदिवासी बहुल है। विकास के मामले में उपेक्षित रहे इस गांव के लोग अब वोट की अहमियत समझने लगे हैैं। विकास के प्रति इनमें जागरूकता का ही असर है कि चुनाव से पहले ही यह अपने प्रतिनिधि की मंशा और सेवा भावना को भांप लेना चाहते हैैं। परीक्षा में सफल घोषित की गई ललिता बिरुवा ने बताया कि वे तीन बार से लगातार कुतरा की सरपंच रही हैं। पहली बार मालूपाड़ा गांव के लोगों ने चुनाव लड़ रहे उम्मीदवारों की परीक्षा ली। ग्रामीणों का फैसला सराहनीय है। उधर, फेल घोषित किए गए नुवास डांग व अन्य प्रत्याशियों ने बताया कि सरपंच उम्मीदवारों का परीक्षा लेना सही निर्णय था, लेकिन पासफेल का रिजल्ट चुनाव से पहले सामने नहीं लाना चाहिए था।गांव में कई समस्याएं मालूपाड़ा गांव में लोग बुनियादी समस्याओं से जूझ रहे हैं। सड़कों की हालत अच्छी नहीं है। वहीं, स्कूलकालेजों में शिक्षक नहीं हैैं। बिजली की भी समस्या रहती है। यहां के लोग बुनियादी सुविधाओं के लिए वर्षों से तरस रहे हैं। कोई इनकी समस्या हल करने में रूचि नहीं ले रहा है। ग्रामीण अपने जनप्रतिनिधियों से ऊब चुके हैं।समस्याओं के निदान के लिए ग्रामीणों ने बनाई है कमेटी मालूपाड़ा के ग्रामीण अपने अधिकारों को लेकर जागरूक हुए हैं। ग्रामीणों ने मिलकर यहां एक पाढ़ा घर कमेटी का गठन किया है। कमेटी की हर सप्ताह बैठक होती है, जिसमें गांव की समस्याओं पर विचारविमर्श कर उसका हल निकालने की कोशिश होती है। गांव की समस्याओं के समाधान के लिए सरपंच की भूमिका अहम होने के कारण योग्य सरपंच उम्मीदवार के चयन के लिए पाढ़ा घर कमेटी ने ही उम्मीदवारों की परीक्षा लेने का निर्णय लिया था। पाढ़ा घर के दीवान प्रदीप लकड़ा के नेतृत्व में ग्रामीणों की उपस्थिति में परीक्षा का आयोजन किया गया था।प्रत्याशियों से पूछे गए प्रश्न आपके सरपंच प्रत्याशी बनने के पांच उद्देश्य क्या हैं ? सरपंच बनने पर आप पांच साल में क्याक्या करेंगे ? चुनाव से पहले किए गए पांच जनसेवा कार्य बताएं ? चुनाव जीतने के बाद क्या इस तरह घरघर घूमेंगे, जैसा वोट मांगने के लिए घूम रहे हैैं? आप अपनी पंचायत को कैसा बनाने का सपना देखते हैं ? आपके पंचायत में वार्ड कितने हैैं और जनसंख्या कितनी है? इससे पहले के सरपंच के बारे में आप क्या जानते हैं? | hindi |
പാചകവാതകം ചോര്ന്ന് സിലിണ്ടറിനു തീപിടിച്ചു അമ്ബലപ്പുഴ : അടുക്കളയില് പാചകവാതകം ചോര്ന്ന് സിലിണ്ടറിനു തീപിടിച്ചു. അമ്ബലപ്പുഴ വടക്ക് ഗ്രാമപ്പഞ്ചായത്ത് 12ാം വാര്ഡില് വളഞ്ഞവഴി എ.ഡി.എഫ്. ജങ്ഷനില് കാട്ടുംപുറം വെളിയില് അബ്ദുല് അസീസിന്റെ വീട്ടില് കഴിഞ്ഞ ഉച്ചക്ക് 12.45 നായിരുന്നു സംഭവം. അഗ്നിബാധയെ തുടര്ന്ന് തീയും പുകയും ഉയരുന്നത് വിവരം ആലപ്പുഴ അഗ്നിരക്ഷാനിലയത്തിലും അമ്ബലപ്പുഴ പോലീസിലും അറിയിച്ചു. നാട്ടുകാരും പരിസരവാസികളും രക്ഷാപ്രവര്ത്തനം ആരംഭിച്ചു. ഗ്രേഡ് അസിസ്റ്റന്റ് സ്റ്റേഷന് ഫയര് ഓഫീസര് ജയസിംഹന്റെ നേതൃത്വത്തില് എത്തിയ അഗ്നിരക്ഷാസേന നാട്ടുകാരോടൊപ്പം ചേര്ന്ന് 15 മിനിറ്റത്തെ ശ്രമഫലം കൊണ്ട് തീപൂര്ണമായി നിയന്ത്രണ വിധേയമാക്കി . വാതകചോര്ച്ച പരിഹരിച്ച് വന്ദുരന്തം ഒഴിവാക്കി. എസ്.ഐ. ചിത്തരഞ്ജന്റെ നേതൃത്വത്തില് അമ്ബലപ്പുഴ പോലീസ് സംഘവും സംഭവസ്ഥലത്ത് എത്തിയിരുന്നു. ആന്റണി, രാജേഷ്, സന്തോഷ്, ഷൈജു, സീനിയര് ഫയര് മെക്കാനിക് കബീര് എന്നിവരും അഗ്നിരക്ഷാസേനാസംഘത്തില് ഉണ്ടായിരുന്നു.The post പാചകവാതകം ചോര്ന്ന് സിലിണ്ടറിനു തീപിടിച്ചു first appeared on MalayalamExpressOnline. | malyali |
हल्द्वानी के स्टेडियम में बनाया कोविड अस्पताल, मरीज आए नहीं और खिलाड़ियों की भी नो एंट्री! हल्द्वानी शहर के इकलौते मिनी स्टेडियम में बनाए गए कोविड अस्पताल Covid 19 Hospital in Haldwani में न तो मरीज आ रहे हैं, न ही अभ्यास के लिए खिलाड़ी आ रहे हैं. कोरोना की दूसरी लहर के दौरान स्टेडियम में 150 बेड का अस्थायी अस्पताल बनाया गया था.यहां कोविड से संबधित सभी नए अपडेट पढ़ें अस्पताल में पड़े बेड अब धूल फांक रहे हैं. अस्पताल बनने के बाद यहां एक भी मरीज भर्ती नहीं हुआ है.7 मई, 2021 को प्रशासन ने स्पोर्ट्स स्टेडियम में इंडोर गेम सेक्शन में कोविड मरीजों के लिए अस्थायी अस्पताल बनाया था. जिसके बाद यहां खिलाड़ियों का आना पूर्ण रूप से बंद कर दिया गया. इससे पहले जब अस्पताल बनाने का कार्य चल रहा था, तो कुछ खिलाड़ी यहां आकर अभ्यास कर लेते थे लेकिन जैसे ही अस्पताल में बेड लगा दिए गए, तो उनका आना पूरी तरह से बंद हो गया.यहां बैडमिंटन, टेबल टेनिस और जिम के लिए बनाए गए हॉल में कोविड मरीजों के लिए बेड लगा दिए गए हैं. इसके चलते बैडमिंटन और टेबल टेनिस खेलने वालों खिलाड़ी अब अभ्यास तक नहीं कर पा रहे हैं.बताते चलें कि 7 मई, 2021 को स्थापित हुए इस 150 बेड के अस्पताल में ऑक्सीजन बेड के अलावा आईसीयू और वेंटिलेटर की सुविधा भी उपलब्ध है. ऑक्सीजन सप्लाई के लिए यहां एक जंबो ऑक्सीजन सिलेंडर भी स्थापित किया गया है. | hindi |
# Circaea kawakamii Hayata SPECIES
#### Status
ACCEPTED
#### According to
International Plant Names Index
#### Published in
null
#### Original name
null
### Remarks
null | code |
ఒడిశా సీఎంతో భేటీ కానున్న జగన్..పలు కీలక అంశాలపై చర్చ.. ఏపీ సీఎం జగన్ భువనేశ్వర్ కి చేరుకున్నారు. కొటియా గ్రామాల వివాదంపై జగన్ ఫోకస్ పెట్టనున్నారు. నేరడి బ్యారేజ్,జంఝావతి ప్రాజెక్టుపై సీఎంలు చర్చించనున్నారు. కాసేపట్లో ఒడిశా సీఎం నవీన్ పట్నాయక్తో జగన్ భేటీ కానున్నారు. ఉభయరాష్ట్రాలకు సంబంధించిన పలు కీలక అంశాలపై చర్చించనున్నారు. వంశధారపై నేరడి దగ్గర బ్యారేజ్ నిర్మాణం, జంఝావతి ప్రాజెక్ట్, కొఠియా గ్రామాల సమస్యపై ప్రధానంగా చర్చించనున్నట్లు తెలుస్తోంది. బ్యారేజీ నిర్మాణానికి ఒడిశా నుంచి 103 ఎకరాలు అవసరమని అధికారులు తెలిపారు. నేరడి బ్యారేజీ నిర్మాణం వల్ల ఒడిశాలో 6 వేల ఎకరాల భూమికి సాగునీరు వస్తుందని అధికారులు వెల్లడించారు. రాష్ట్రాల మధ్య సమస్యలపై చర్చించనున్నారు. | telegu |
Sophie Choudry: నెవర్ బిఫోర్.. ఎవర్ ఆఫ్టర్ అనేలా సోఫీ అందాలు 1 నేనొక్కడినే చిత్రంలో లండన్ బాబు.. లండన్ బాబు అంటూ స్పెషల్ సాంగ్లో చిందులేసిన సోఫీ చౌదరి బహుముఖ ప్రజ్ఞాశాలి. నటిగా, యాంకర్గా, డ్యాన్సర్గా అదరగొడుతూ ఉంటుంది. బిగ్ బాస్ సీజన్ 8లోను సందడి చేసి అశేష ప్రేక్షకాదరణ పొందింది. అయితే పెద్దగా సినిమాల్లో అవకాశాలు రానప్పటికీ.. సోషల్మీడియా ద్వారా మాత్రం తన అందాలతో హీటెక్కిస్తుంది సోఫీ. తాజాగా లోదుస్తుల్లో.. అందాల విందు ఇస్తూ కొన్ని పిక్స్ షేర్ చేసింది ఈ బ్యూటీ. మాల్దీవుల్లోఈ ముద్దుగుమ్మ బికినీలోఅందాలు ఆరబోస్తూ కేక పెట్టిస్తుంది. సోఫీని ఇలా చూసిన అబ్బాయిల గుండెల్లో రైళ్లు పరిగెడుతున్నాయి. నెవర్ బిఫోర్, ఎవర్ ఆఫ్టర్ అనేలా సోఫీ అందచందాల ప్రదర్శన ఉంది. సోఫి చౌదరీ ఇటీవల కాలంలో ఎక్కువ శాతం మాల్దీవుల్లోనే గడుపుతోంది. బోట్ రెయిడ్స్ను ఎంజాయ్ చేస్తోంది. సముద్రంలో డాల్ఫిన్లను ఛేజ్ చేస్తూ కాలం గడిపేస్తోంది. సోఫీ.. ఎంటీవీలో లవ్లైన్, జలక్ దిక్లా జా 7 లాంటి కార్యక్రమాలకు హోస్ట్గా చేసింది. ప్యార్ కే సైడ్ ఎఫెక్ట్స్, మనీ హై తో హనీ హై, ఆ దేకే జరా, వన్స్ అపాన్ ఏ టైమ్ ఇన్ ముంబై దొబారా లాంటి చిత్రాల్లో ఆమె నటించింది. | telegu |
এই প্রতিকারটি অনুসরণ করুন, বিবাহিত জীবনে কখনও মধুরতা শেষ হবে না দাম্পত্য জীবন যদি ভাল চলে তবে সবকিছুই ভাল থাকবে তবে বিবাহিত জীবনে অনেক সময় এমন কিছু পরিস্থিতি দেখা যায় যে ব্যক্তি বিচলিত হন এ কারণে ব্যক্তির আচরণগত জীবন এবং সামাজিক জীবনও প্রভাবিত হয় বিবাহিত জীবনে তিক্ততার কারণে অনেক সময় সম্পর্ক বিচ্ছেদের পথে যায় যে কোনও পরিবারে সমৃদ্ধির প্রথম ধাপটি হলো ঘরে শান্তি জ্যোতিষশাস্ত্র তে, এই জাতীয় অনেকগুলি পদক্ষেপের বর্ণনা দেওয়া হয়েছে, যা বাড়ি থেকে দুঃখকষ্ট দূর করতে পারে সূর্যোদয় এর আগে স্নান করুন এর পরে যে কোনও শিব মন্দিরে যান শিবলিঙ্গকে জল অর্পণ করার সময় এই মন্ত্রটি পুরো নিষ্ঠার সাথে জপ করুন ওম নমঃ সম্ভায়া চ মায়ো ভাবায় চ নাম: শঙ্করাই চ মায়স্করাই চ নম: শিবায়া চ শিবতরায় চ বৃহস্পতিবার স্বামীস্ত্রী একসাথে বসে ভগবান বৃহস্পতির উপাসনা করেন বলা হয়ে থাকে যে এটি করার মাধ্যমে সম্পর্কের মধ্যে সর্বদা মাধুরী থাকে, মনে রাখবেন যে পূজাতে বসার আগে স্বামীস্ত্রী উভয়েরই হলুদ রঙের পোশাক পরা উচিত স্বামী যদি সর্বদা অসন্তুষ্ট থাকে, স্ত্রীর কথায় মনোযোগ না দেয় এবং যার কারণে দাম্পত্য কলহ ঘটে চলেছে এবং সমস্ত প্রচেষ্টা ব্যর্থ হচ্ছে, তবে ভগবান শঙ্কর ও দেবী পার্বতীর ধ্যান করার সাথে সাথেই সোমবার থেকে নিম্নলিখিত মন্ত্রের মালা জপ করুন মন্ত্রঃ ওঁ ক্লেইন ত্রিম্বকাম যজ্ঞামে সুসুধীন পতিস্তেনম্ উর্বরুকামিভ বন্ধনাদিতো মুসিল মমরিত ক্লি আন পরিবারে সুখ, শান্তি এবং সমৃদ্ধির জন্য, প্রতিদিন প্রথম রুটির চারটি সমান অংশ করুন একটি গরুকে, অন্যটি কালো কুকুরকে, তৃতীয়টি কাককে দিন এবং চতুর্থ অংশটি ক্রুশে রাখুন মন্দিরে অশোক গাছের সাতটি পাতা রেখে পূজা করুন এগুলি শুকোতে শুরু করলে, নতুন পাতা রাখুন এবং পুরাতনগুলি পিপল গাছের নীচে রাখুন এ কারণে ঘরে সুখ ও শান্তি বজায় থাকবে আপনার সমস্যা একটি ঘুড়িতে লিখুন এটি বাতাসে ছেড়ে দিন এটি অবিচ্ছিন্নভাবে ৭ দিন করুন সমস্ত সমস্যা দূর হবে এবং ঘরে সুখ ও শান্তি থাকবে যদি স্বামীস্ত্রীর মধ্যে প্রায়শই কোনও লড়াই বা মতবিরোধ দেখা দেয় তবে বৃহস্পতিবার ভগবান বিষ্ণুর পূজা করুন এবং একটি লাল সুতো বা মলি অর্পণ করুন পূজার পরে, এই মলিটি আপনার ডান কব্জির উপর বেঁধে রাখুন এমনটা করলে সম্পর্কের মধুরতা বাড়বে | bengali |
\begin{document}
\title{Optimal scaling quantum linear systems solver via discrete adiabatic theorem}
\date{\today}
\author{Pedro C.~S.~Costa} \MQ
\author{Dong An} \UMD \Google
\author{Yuval R.~Sanders} \MQ \UTS
\author{Yuan Su} \Google
\author{Ryan Babbush} \Google
\author{Dominic W.~Berry} \MQ
\begin{abstract}
Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian. Such approaches enabled near-linear scaling in the condition number $\kappa$ of the linear system, without requiring a complicated variable-time amplitude amplification procedure. However, the most efficient of those procedures is still asymptotically sub-optimal by a factor of $\log(\kappa)$. Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete time evolutions.
We use this discrete adiabatic theorem to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in $\kappa$, matching a known lower bound on the complexity.
Our $\mathcal{O}(\kappa\log(1/\epsilon))$ complexity is also optimal in terms of the combined scaling in $\kappa$ and the precision $\epsilon$.
Compared to existing suboptimal methods, our algorithm is simpler and easier to implement.
Moreover, we determine the constant factors in the algorithm, which would be suitable for determining the complexity in terms of gate counts for specific applications.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
Finding the solution to a system of linear equations is a fundamental task that underlies many areas of science and technology. A classical linear system solver takes time proportional to the number of unknown variables even to write down the solution, and thus has a prohibitive computational cost for solving large linear systems. However, a quantum computer with a suitable input access can produce a quantum state that encodes the problem solution much faster than its classical counterpart. The first quantum algorithm for the quantum linear system problem (QLSP) was proposed by Harrow, Hassidim, and Lloyd (HHL) \cite{Harrow_2009}, and has been subsequently refined by later work. Due to the ubiquitous nature of the problem, quantum algorithms for QLSP have found a variety of applications, such as computing electromagnetic scattering \cite{Clader13}, solving differential equations \cite{BerryJPA14,BerryCMP17}, data fitting \cite{Wiebe2012}, machine learning \cite{supvec,lloyd2013quantum}, and more general solution of partial differential equations \cite{Ashley2016}.
Specifically, the goal of QLSP is to produce a quantum state $\ket{x}$ proportional to the solution of linear system $Ax=b$, where $A$ is an $N$-by-$N$ matrix.
The complexity of solving QLSP depends on various input parameters, such as the problem size $N$, the sparsity (for sparse linear systems), the norm of the coefficient matrix $A$, the condition number $\kappa$, and the error $\epsilon$ in the solution.
To simplify the discussion, we assume that $\norm{A}=1$ and hence $\norm{A^{-1}}=\kappa$, where $\norm{\cdot}$ denotes the spectral norm.
To further simplify the analysis, we assume that we have a block encoding of the coefficient matrix $A$ and a given operation to prepare the target vector $\ket{b}$, and consider the number of queries to these oracles.
One can also consider the complexity in terms of the number of calls to entries of a sparse matrix, as in \cite{Harrow_2009}, but there are standard methods to block encode sparse matrices \cite{rootd}, so our result can be easily applied to that case.
These simplifications mean that the only relevant parameters which our algorithm depends on are $\kappa$ and $\epsilon$.
The original algorithm proposed by HHL has a complexity scaling quadratically with the condition number $\kappa$ and linearly with the inverse accuracy $1/\epsilon$ \cite{Harrow_2009}. The scaling with the condition number was improved by Ambainis using ``variable time amplitude amplification'' \cite{ambainis2010variable}; the resulting algorithm has a near-linear dependence on $\kappa$ but a much worse dependence on $1/\epsilon$. A further improvement was provided in \cite{CKS}, which yields a complexity logarithmic in the allowable error $\epsilon$.
Unfortunately, algorithms based on variable time amplitude amplification \cite{ambainis2010variable,CKS} perform multiple rounds of recursive amplitude amplifications and can be challenging to implement in practice.
To address this, recent work has suggested alternative approaches based on adiabatic quantum computing (AQC). AQC is a universal model of quantum computing that has been shown to be polynomially equivalent to the standard gate model~\cite{farhi2000quantum,Aharonov2007}.
In AQC one encodes the solution to a computational problem in the ground state of a Hamiltonian $H_1$. Then, one initializes a quantum system in the ground state of an easy-to-prepare Hamiltonian $H_0$ and slowly deforms from the ground state of $H_0$ to the ground state of $H_1$ under a time-dependent Hamiltonian that interpolates between the two, such as $H(s)=(1-s)H_0+sH_1$. The advantage of using the adiabatic approach to solve QLSP as in \cite{PhysRevLett.122.060504} is that it naturally provides complexity close to linear in $\kappa$, without the highly complicated variable time amplitude amplification procedure. That work was further improved in \cite{an2019quantum} then \cite{Lin2020optimalpolynomial}, which gave complexity logarithmic in $\epsilon$ by using eigenstate filtering. We summarize key developments reducing the complexity in \tab{linear_systems_history}.
\begin{table*}[ht]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Year
& Reference
& Primary innovation
& Query complexity\\
\hline\hline
2008
& Harrow, Hassidim, Lloyd \cite{Harrow_2009}
& first quantum approach
& ${\cal O}(\kappa^2 / \epsilon)$\\
2012
& Ambainis \cite{ambainis2010variable}
& variable-time amplitude amplification
& ${\cal O}(\kappa (\log (\kappa) /\epsilon)^3)$\\
2017
& Childs, Kothari, Somma \cite{CKS}
& Fourier/Chebyshev fitting using LCU
& ${\cal O}(\kappa\, {\rm polylog} (\kappa/\epsilon))$\\
2018
& Subasi, Somma, Orsucci \cite{PhysRevLett.122.060504}
& adiabatic randomization method
& ${\cal O}((\kappa \log \kappa) / \epsilon)$\\
2019
& An, Lin \cite{an2019quantum}
& time-optimal adiabatic method
& ${\cal O}(\kappa \, {\rm polylog}(\kappa/\epsilon)) $\\
2019
& Tong, Lin \cite{Lin2020optimalpolynomial}
& Zeno eigenstate filtering
& ${\cal O}(\kappa \log (\kappa/\epsilon))$\\
2021
& this paper
& discrete adiabatic theorem
& ${\cal O}(\kappa \log(1/\epsilon))$\\
\hline
\end{tabular}
\caption{\label{tab:linear_systems_history} History of the lowest scaling algorithms for solving linear systems of equations on a quantum computer. Specifically, the problem is to prepare the state $\ket{x}$ given oracular access to the matrix $A$ and the ability to prepare the initial state $\ket{b}$ encoding a vector $b$ with the relation $A x = b$. Here, $\kappa$ is the condition number of $A$ and $\epsilon$ is the target precision to which we would like to prepare the state $\ket{x}$. However, the cost of a query for all classical algorithms is expected to scale polynomially in $N$ (the dimension of the matrix $A$), whereas on a quantum computer it is possible to make queries in complexity scaling as ${\cal O}(\textrm{polylog}(N))$ when $A$ is a sparse matrix. Query complexity of $\Omega(\kappa \log(1/\epsilon))$ is a known lower bound on the complexity.}
\end{table*}
It is known that a quantum algorithm must make at least $\Omega(\kappa \log(1/\epsilon))$ queries in general to solve the sparse QLSP problem \cite{RobinAram}. Therefore, the method in \cite{Lin2020optimalpolynomial} is already optimal in the scaling with solution accuracy $\epsilon$. However, a question left open was: how can we achieve an optimal scaling with the condition number $\kappa$, or is it possible to prove a lower bound ruling out this improvement? From the algorithmic perspective, finding a quantum algorithm with linear $\kappa$-scaling is technically challenging. Previous fast linear system solvers depend on polynomial approximations to implement the inverse function $1/x$ on $x\in[1/\kappa,1]$ \cite{CKS}, or truncations of the Dyson series to implement the continuous adiabatic evolution \cite{PhysRevLett.122.060504,an2019quantum,Lin2020optimalpolynomial}. In either case, an extra $\mathrm{polylog}(\kappa)$ factor is required to suppress the truncation or approximation error, resulting in a superlinear scaling with the condition number.
In this work, we develop a quantum algorithm for solving systems of linear equations with complexity $\mathcal{O}(\kappa\log(1/\epsilon))$. That is, we achieve a strictly linear scaling with $\kappa$, while maintaining the logarithmic scaling with $1/\epsilon$ from the best previous algorithms. Combining with the lower bound of \cite{RobinAram}, we establish for the first time a quantum linear system algorithm with optimal scaling in the condition number.
It is also optimal in the combined scaling with $\kappa$ and $\epsilon$, because one cannot for example reduce the scaling to $\mathcal{O}(\kappa+\log(1/\epsilon))$.
We formally state our result in \sec{filter} and preview it here.
\begin{customthm}{}[QLSP with linear dependence on $\kappa$]
Let $Ax=b$ be a system of linear equations, where $A$ is an $N$-by-$N$ matrix with $\norm{A}=1$ and $\norm{A^{-1}}=\kappa$. Given an oracle block encoding the operator $A$ and an oracle preparing $\ket{b}$, there exists a quantum algorithm which produces the normalized state $\ket{A^{-1}b}$ to within error $\epsilon$ using a number
\begin{equation}
\mathcal{O}(\kappa\log(1/\epsilon))
\end{equation}
of oracle calls.
\end{customthm}
Our algorithm is conceptually simple and easy to describe.
All it requires is a sequence of quantum walk steps corresponding to a qubitised form of the Hamiltonian used in prior work.
It completely avoids the heavy mechanisms of variable time amplitude amplification or the truncated Dyson-series subroutine from previous methods.
Moreover, we provide a bound on the constant prefactor for our approach, that will allow estimation of the complexity in terms of the number of gates for specific applications.
We expect that our estimate of the prefactor can be tightened, and our algorithm will be the most efficient for the early fault-tolerant regime of quantum computation as well as having the best asymptotic scaling for large-scale applications.
The new insight that allows us to establish the optimal $\kappa$-scaling is the use of a discrete quantum adiabatic theorem, a result proved by Dranov, Kellendonk, and Seiler (DKS)~\cite{DKS98}. Unlike the continuous version, the discrete adiabatic theorem is formulated based on a quantum walk operator $W_T(s)$. Provided that the steps of quantum walk vary sufficiently slowly, the eigenstates of the walk operator can be approximately preserved throughout the entire discrete adiabatic evolution. Indeed, DKS showed that the error in the evolution scales as $\mathcal{O}(1/T)$ for $T$ steps of the walk. However, their analysis overlooked the scaling with other parameters, in particular, the spectral gap dependence.
In the case of solving QLSP, the gap depends on $\kappa$, so the result of DKS is not sufficient to give the $\kappa$-dependence of the algorithm.
Here, we give a complete analysis of the discrete adiabatic theorem, keeping track of all the parameters of interest while fixing several minor mistakes in the original proof.
In developing our quantum linear system algorithm, we provide an improved method of filtering the final state which may be of independent interest.
Prior methods were based on singular value processing \cite{Lin2020optimalpolynomial}, which requires a sequence of rotations to be found by a numerically demanding procedure~\cite{Childs18,Haah2019product,Dong21,Chao2020}.
Our method is just as efficient, but the sequence of operations needed is easily determined.
Combining the discrete adiabatic theorem on the qubitised quantum walk with the improved eigenstate filtering then gives our result on the solution of linear systems.
The remainder of the paper is organized as follows.
In the following we give more detailed background and summarise our result in \sec{summary}.
Then in \sec{adtheo} we give a thorough proof of the discrete adiabatic theorem.
We base our method on the approach of DKS, but make many of the details rigorous and provide a strict bound on the error including constant factors.
We apply the discrete adiabatic theorem to the QLSP in \sec{linsys}.
In \sec{filter} we provide our general method of filtering that is just as efficient as that based on singular value processing.
\section{Discrete adiabatic theorems}
\label{sec:summary}
\subsection{Background}
\label{sec:summary_background}
Before presenting our results, let us present the main ideas of the DKS bound on the error in discrete-time adiabatic evolution \cite{DKS98}.
In this proposal the model of the adiabatic evolution is based on a sequence of $T$ walk operators $\{W_T(n/T): n \in \mathbb{N}, 0\leq n \leq T-1\}$.
That is, the system is initially prepared in a state $\ket{\psi_0}$, then the sequence of unitary transformations $W_T(n/T)$ have the effect
$\ket{\psi_0} \mapsto \ket{\psi_1} \mapsto \cdots$.
To model this evolution, with $s := n/T$ we can write
\begin{equation}
\label{eq:U_as_product_of_W}
U_T(s) = \prod_{n = 1}^{sT-1} W_T\left(n/T\right),
\end{equation}
and $U_T(0) \equiv \mathds{1}$, such that $\ket{\psi_n} = U_T(s)\ket{\psi_0}$.
The adiabatic limit is then the limit $T \to \infty$.
Alternatively we can construct the total unitary evolution recursively as
\begin{equation}
\label{eq:discrete_time_schroedinger_eqn_rescaled}
U_T (s + 1/T) = W_T (s+1/T) U_T (s), \quad U_T (0) = \mathds{1}.
\end{equation}
The adiabatic limit is then the limit $T \to \infty$.
For the purpose of quantum algorithm design,
we are trying to choose $U_T$ so that
$\lim_{T \to \infty} U_T (1) \ket{\psi_0} = \ket{\psi_{\rm target}}$,
where $\ket{\psi_{\rm target}}$ is a desired `target' state that
enables us to solve a computational problem.
In order for this to be an accurate adiabatic evolution yielding the target state,
$U(n/T) \ket{\psi_0}$ should be approximately an eigenstate of $W (n/T)$ for all $n$.
We need to establish some terminology before we can present the statement of the result from \cite{DKS98}.
For each $T \in \mathbb{R}$ and $n \in \mathbb{N}$, introduce a projector $P_T (s)$ (with $s \equiv n/T$ as before) called the
\textit{spectral projection}, which projects onto the eigenspace of interest.
In addition, the \textit{complementary spectral projection} $Q_T (s)=\openone-P_T (s)$ projects onto all orthogonal eigenvectors.
An operator representing the ideal adiabatic evolution is denoted $U_T^A(s)$, so that
\begin{equation}
P_T(s)=U_T^A(s) P_T (0) U_T^{A\dagger}(s) .
\end{equation}
That is, evolving the original eigenspace to step $n=sT$ under the ideal adiabatic evolution gives the corresponding eigenspace for the walk operator $W_T(s)$.
The adiabatic theorem is a statement about how close the ideal adiabatic evolution $U_T^A(s)$ is to the real evolution $U_{T}(s)$ at a given time.
Beginning with the initial state $\ket{\phi(0)}$ in the subspace of interest so $(P_T(0)\ket{\phi(0)}=\ket{\psi(0)})$, the goal is to bound the \textit{error} between $U_{T}$ and its ideal evolution $U_T^A$ by an expression of the form
\begin{equation}
\label{eq:goal}
\left\|\left(U_T(s)-U_T^A(s)\right)\ket{\phi(0)}\right\| \leq \left\|U_T(s)-U_T^A(s)\right\| \leq \frac{\theta}{T} ,
\end{equation}
where $\| \cdot \|$ is the spectral norm.
Proving this result shows that increasing the number of steps reduces the error.
The constant $\theta$ in \cref{eq:goal} is a constant independent of the total time $T$, but depends on the gap $\Delta(s)$ between the eigenspace of interest and the complementary eigenspace.
In \cite{DKS98} it was shown that the error is $\mathcal{O}(1/T)$, which means that there exists some constant $\theta$, but that constant and its dependence on the gap were not determined.
That is a crucial difficulty in applying the result to the QLSP,
because the gap in using the adiabatic approach to the QLSP depends on the condition number $\kappa$.
Therefore, to determine the complexity of the algorithm in terms of $\kappa$, we need to know the dependence of the error on the gap.
In particular, we will show that the error scales as $\mathcal{O}(\kappa/T)$, which means that to obtain the solution to fixed error one can use $T=\mathcal{O}(\kappa)$ steps.
Then complexity linear in $\kappa$ and logarithmic in $1/\epsilon$ can be obtained using filtering.
To show this result we cannot simply use the result as given \cite{DKS98}, and need to derive the bound for the error far more carefully in order to give the dependence on the gap.
\subsection{Our result}
\label{sec:summary_result}
Our main goal here is to provide the explicit dependence on the gap in the discrete adiabatic theorem in order to improve the version given in \cite{DKS98}.
The starting point is to replace the general order scaling
\begin{equation}
W_T\left(s+1/T\right)-W_T\left(s\right)\approx \mathcal{O}(T^{-1}),
\end{equation}
with an upper bound with explicit schedule dependence,
\begin{equation}
\label{eq:main_ass}
\left\|W_T\left(s+1/T\right)-W_T\left(s\right)\right\|\leq \frac{c(s)}{T}.
\end{equation}
Implicit in this definition is the assumption that the behaviour of $W_T (s)$ is sufficiently smooth that $c(s)$ can be chosen independently of $T$.
This will need to be shown for the given applications.
More generally, we will need to consider higher-order differences, which result in values of $c_k(s)$ given in the following definition.
\begin{definition}[Multistep Differences]
\label{def:difs}
For a positive integer $k$, the $k^{\rm th}$ difference of $W_T$ is
\begin{equation}
D^{(k)} W_T(s) :=
D^{(k-1)} W_T \left( s + \frac{1}{T} \right) - D^{(k-1)} W_T(s), \quad
D^{(1)} W_T(s) := DW_T(s) =
W_T \left( s + \frac{1}{T} \right) - W_T (s).
\end{equation}
For $T > 0$,
we define the function $c_k(s)$, which is implicit dependent of $T$, such that
\begin{equation}\label{eqn:assump1}
\left\|D^{(k)} W_T(s)\right\| \leq \frac{c_k(s)}{T^{k}}.
\end{equation}
We then define the $\hat{c}_k(s)$ taking into account neighbouring steps as
\begin{equation}
\label{eq:chat}
\hat{c}_k(s) = \max_{s' \in \left\{s-1/T,s,s+1/T\right\} \cap [0,1-k/T] } c_k(s') .
\end{equation}
\end{definition}
The principle of the gap is that it separates the eigenvalues of $W_T(s)$ into two groups that depend on the time parameter $s$.
Since $W_T(s)$ is unitary, these are groups on the unit circle in the complex plane.
Because it is on the unit circle, we need to separate these groups of eigenvalues with gaps in two locations.
We will denote one set of eigenvalues as $\sigma_P(s)$ and the other as $\sigma_Q(s)$, with corresponding projectors $P_T(s)$ and $Q_T(s)$, respectively.
We also need to account for the gaps for successive operators $W_T(s)$ and $W_T(s+1/T)$.
That is, there needs to be a gap between $\sigma_P(s)\cup \sigma_P(s+1/T)$ and $\sigma_Q(s)\cup \sigma_Q(s+1/T)$.
Moreover, we need to ensure that these regions are non-interleaved.
To do this, we will define arcs that contain the eigenvalues, such that
\begin{equation}
\sigma^{(1)}_P \supseteq \sigma_P(s)\cup \sigma_P(s+1/T), \qquad \sigma^{(1)}_Q \supseteq \sigma_Q(s)\cup \sigma_Q(s+1/T).
\end{equation}
To rule out interleaved regions, these arcs cannot intersect, and we consider the gap between these arcs.
We are interested in the case where this only has a small effect on the gap.
In turn this means that $T$ should not be too large, so we introduce a lower bound $T^*$ on the values of $T$ allowed.
We therefore define the multistep gaps as follows.
\begin{definition}[Multistep Gap]
\label{def:gaps}
For $T \in \mathbb{N}$ and $k$ a non-negative integer, $\Delta_k(s)$ is defined to be the minimum angular distance between arcs $\sigma^{(k)}_P$ and $\sigma^{(k)}_Q$, which satisfy
\begin{equation}\label{eqn:assump2}
\sigma^{(k)}_P \supseteq \bigcup_{l=0}^k \sigma_P(s+l/T), \qquad
\sigma^{(k)}_Q \supseteq \bigcup_{l=0}^k \sigma_Q(s+l/T),
\end{equation}
for $T\ge T_*$.
The gap $\Delta(s)$, which is also implicitly dependent on $T$, is then in most cases the minimum gap for three successive steps, except in the cases at the boundaries:
\begin{equation}\label{eq:minGaps}
\Delta(s) = \begin{cases}
\Delta_2(s), & 0 \leq s \leq 1-2/T, \\
\Delta_1(s), & s = 1-1/T, \\
\Delta_0(s), & s = 1.
\end{cases}
\end{equation}
Finally, $\check{\Delta}(s)$ is an adjustment for $\Delta(s)$ at neighbouring points:
\begin{equation}
\label{eq:fhat}
\check{\Delta}(s) = \min_{s' \in \left\{s-1/T,s,s+1/T\right\} \cap [0,1] } \Delta(s').
\end{equation}
\end{definition}
Note that we have freedom to choose larger arcs than necessary, so these are lower bounds on the gap, though we will often call them the ``gap'' for convenience.
Also, given $\Delta_2(s)$, one can always choose arcs $\sigma^{(k)}_P$ and $\sigma^{(k)}_Q$ for $k=0,1$ such that $\Delta_k(s)\ge \Delta_2(s)$.
This means that in \cref{eq:minGaps} we can simply use $\Delta_2(s)$, rather than taking the minimum of $\Delta_k(s)$ for $k \in \{0,1,2\}$.
We have proven two forms of the discrete adiabatic theorem.
One is highly complicated, so we give it explicitly in later in \cref{sec:dat1}.
Here we instead give a simplified but looser form of the discrete adiabatic theorem.
\begin{theorem}[The Second Discrete Adiabatic Theorem]\label{cor:adia}
Suppose that the operators $W_T (s)$ satisfy
$\left\|D^{(k)}W_T(s)\right\|\leq c_k(s)/T^k$ for $k = 1,2$, as per \cref{eqn:assump1}, and $T \geq \max_{s\in [0,1]} (4\hat{c}_1(s)/\check{\Delta}(s))$,
Then for any time $s$, s.t., $sT \in \mathbb{N}$, we have
\begin{align}
\quad \|U_T(s) - U_T^{A}(s)\|
& \leq \frac{12\hat{c}_1(0)}{T\check{\Delta}(0)^2}+ \frac{12\hat{c}_1(s)}{T\check{\Delta}(s)^2} + \frac{6\hat{c}_1(s)}{T\check{\Delta}(s)} + 305\sum_{n=1}^{sT-1}\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} \nonumber \\
& \quad + 44\sum_{n=0}^{sT-1} \frac{ \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^2} + 32\sum_{n=1}^{sT-1}\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2},
\end{align}
where $\hat{c}_k(s)$ and $\check{\Delta}(s)$ are defined in \cref{def:difs} and \cref{def:gaps}, respectively.
\end{theorem}
Note that this theorem depends on the first and second differences, described by $\hat{c}_1(s)$ and $\hat{c}_2(s)$, respectively.
These are analogous to the first and second derivatives in the continuous form of the adiabatic theorem, so we can see that these results are analogous.
We have three single terms with $1/T$ scaling, and three sums with $1/T^2$ scaling, which gives overall scaling of the complexity as $\mathcal{O}(1/T)$.
We also have a cubic dependence in the inverse gap $1/\Delta$ in the first sum given.
In choosing the quantum walk, one would aim to schedule the variation of $W_T$ such that they vary more slowly where the gap is small, making $\hat{c}_1$ smaller.
\section{The first adiabatic theorem}
\label{sec:adtheo}
In this section, we prove our first form of the discrete adiabatic theorem, given later as \cref{theoAdia}, and then use it to prove \cref{cor:adia}. Following the general method and notation of \cite{DKS98}, we use the \textit{wave operator}
\begin{equation}
\label{eq:waveOp}
\Omega_T (s) := U^{A\dagger}_T(s) U_T (s) .
\end{equation}
The aim of the discrete adiabatic theorem, the first and the second, is to prove that $\Omega_T (s)$ is close to the identity because
\begin{equation}
\left\|U_T(s)-U_T^A(s)\right\| = \left\|U_{T}^{A\dagger}(s)U_T(s)-\mathds{1}\right\| = \left\|\Omega_T (s)-\mathds{1}\right\|.
\end{equation}
In \cite{DKS98} it was shown that $\Omega_T (s) = \mathds{1} + \order{1/T}$, but we instead aim to provide the explicit bounds dependent on the gap.
To prove the bound, one can define a \emph{kernel function} $K_T (s)$ as well, which corresponds to the difference of a single step of $\Omega_T (s)$ from the identity.
The wave operator at step $n$ is then given by
\begin{equation}
\label{eq:waveOp_kernel}
\Omega_T(n/T) = I - \frac{1}{T}\sum_{m=0}^{n-1} K_T(m/T)\Omega_T(m/T).
\end{equation}
The goal is then to show that the sum is small.
This is done with a discrete form of the summation by parts formula, giving our first discrete adiabatic theorem.
\subsection{Operator definitions}
\begin{figure}
\caption{Illustration of the choice of the contour $\Gamma_T(s,k)$ for $k = 0$.
That is, we consider the eigenvalues for only a single step of the walk.
The red dots indicate the spectrum of interest, which will often just be a single eigenvalue, for example for a ground state.
The contour around the spectrum of interest is used to obtain a projector onto the spectrum of interest.
For the illustration, we use a contour with radius $2$, but in practice, we take the limit that the radius goes to infinity. }
\label{fig:contour_1}
\end{figure}
We next define the operators that are needed to understand the proof.
Let $\Gamma_T(s)$ be a sector contour enclosing the spectrum of interest, for example see \cref{fig:contour_1}.
Then the spectral projection $P_T(s)$ onto the spectrum of interest is given by the integral
\begin{equation}
\label{eq:ProjectCount}
P_T(s)=\frac{1}{2 \pi i} \oint_{\Gamma_T(s)} R_T(s,z) dz,
\end{equation}
where
\begin{equation}
\label{eq:resolv}
R_T(s,z)\coloneqq\left(W_T(s)-z I\right)^{-1},
\end{equation}
is the resolvent of $W_T(s)$.
Let
\begin{align}\label{eq:STdef}
S_T(s,s')&\coloneqq P_T(s)P_T(s')+Q_T(s)Q_T(s'), \\
v_T(s,s') &\coloneqq \sqrt{S_T(s,s')S^{\dagger}_T(s,s')}
= \sqrt{I-\left(P_T(s)-P_T\left(s'\right)\right)^2},
\end{align}
and
\begin{equation}
\label{eq:V}
V_{T}(s,s') \coloneqq v_T(s,s')^{-1}S_T(s,s'),
\end{equation}
which is the unitary of the left polar decomposition of $S_T(s,s')$ (see Eq.~(11) of \cite{DKS98}).
We use the shorthand notations $S_T(s) = S_T(s+1/T,s)$, $v_T(s) = v_T(s+1/T,s)$, $V_T(s) = V_T(s+1/T,s)$, and define
(see Eqs.~(7) and (10) of \cite{DKS98})
\begin{align}
\label{eq:Wa}
W_T^A(s) & \coloneqq V_T(s)W_T(s), \\
U_T^A\left(s+\frac{1}{T}\right) & \coloneqq W_T^{A}(s)U_T^A(s),\\
U^A_T(0) & \coloneqq I.
\end{align}
It can be checked from the definition that $V_T(s)$ is a unitary operator, and thus $W_T^{A}$ and $U_T^A$ are unitary.
In fact, $W_T^{A}$ is exactly the adiabatic walk operator, and the corresponding $U_T^{A}$ is the corresponding adiabatic evolution operator.
To describe the proof we use the \emph{wave operator}
\begin{equation}\label{eq:Omegadef}
\Omega_T(s) \coloneqq U_T^{A\dagger}(s)U_T(s),
\end{equation}
which describes the difference between the actual evolution given by $U_T(s)$ and the ideal adiabatic evolution $U_T^{A}(s)$.
To demonstrate that the evolution is close to adiabatic, we should have $\Omega_T(s)$ close to $I$.
The \emph{ripple operator} is defined as
\begin{equation}\label{eq:ripple}
\Theta_T (s) \coloneqq \Omega_T (s + 1/T) \Omega_T^\dagger (s),
\end{equation}
so the wave operator is a product of ripple operators for each time step.
The kernel function is defined as
\begin{equation}
\label{eq:K}
K_T(s) \coloneqq T( I - \Theta_T(s)),
\end{equation}
and should be close to zero for the evolution to be close to adiabatic.
\subsection{Sequence of Lemmas}
Next we give an outline of the lemmas that will be proven and how they fit together to provide the final theorem.
In the proof of the discrete adiabatic theorems, we keep with the the definition giving in \cref{eq:waveOp_kernel} for the wave operator.
The goal is then to upper bound the sum to show that $\Omega_T(n/T)$ is close to the identity, and therefore the evolution is close to adiabatic.
In the sum we substitute the identity being equal to the sum of projections onto the desired subspace and the orthogonal subspace.
That gives us four sums.
Two of these are ``diagonal'' sums with two projections onto the same subspace, and two are ``off-diagonal'' sums with two different projections.
The diagonal sums are relatively easily bounded, whereas for the off-diagonal sums are more difficult,
for those we use the ``summation by parts formula'' in \cref{lem:sum_by_parts}.
In that Lemma we use operators $X$ and $Y$, which will be taken to be $T(I-V_T^\dagger)$ and $\Omega_T$.
In \cref{lem:sum_by_parts} we define operators $\tilde X$, $A$, $B$ and $Z$.
The operator, $\tilde X$, is defined as a contour integral including $X$, then we have $B$ and $Z$ defined in terms of $\tilde X$.
The sequence of lemmas used to prove the first adiabatic theorem (\cref{theoAdia}) are listed below.
\begin{itemize}
\item In \cref{lem:P} we bound the norms $\|DP_T\|$ and $\|D^{(2)}P_T\|$.
The quantity $DP_T(s)$ is the difference in $P_T$ at successive time steps, and $D^{(2)}P_T$ is the difference in $DP_T$.
These quantities are bounded in terms of the bounds on $DW_T$ and $D^{(2)}W_T$.
\item \cref{lem:V_v2} gives $V_T$ in terms of $P_T$ and a new operator $\mathcal{F}_T$.
\item \cref{lem:V_bound2} uses $\mathcal{F}_T$ to place an upper bound on the norm of $V_T-I$.
Because $X$ is $T(I-V_T^\dagger)$, that enables us to place an upper bound on $X$.
\item \cref{lem:DV_v2} provides an upper bound on the norm of $D\mathcal{F}_T$, which enables us to place an upper bound on $DV_T$.
This uses the upper bounds on $\|DP_T\|$ and $\|D^{(2)}P_T\|$ from \cref{lem:P}.
\item \cref{lem:WAv3} places an upper bound on $\|DW_T^A\|$ using the upper bound on $DV_T$ from \cref{lem:DV_v2}.
Recall that $W_T^A$ is the ideal step for adiabatic evolution.
\item \cref{lem:DOmega2} places an upper bound on $\|D\Omega_T\|$ using the upper bound on $\|V_T-I\|$ from \cref{lem:V_bound2}.
\item \cref{lem:Xt} places upper bounds on $\|\tilde X\|$ and $\|D\tilde X\|$ in terms of $\|X\|$ and $\|DX\|$.
Recall that $X$ will correspond to $T(I-V^\dagger)$.
\item \cref{lem:ABZ2_v2} places upper bounds on the norms of the $A$, $B$ and $Z$ operators.
The bound on $\|A\|$ uses the bound on $\|V_T-I\|$ from \cref{lem:V_bound2}.
The bounds on $\|B\|$ and $\|Z\|$ use the bounds on $\|\tilde X\|$ and $\|D\tilde X\|$ from \cref{lem:Xt} as well as the bound on $\|DW_T^A\|$ from \cref{lem:WAv3}.
\end{itemize}
Finally, the bounds on $\|A\|$, $\|B\|$ and $\|Z\|$ from \cref{lem:ABZ2_v2} are used in the summation by parts formula in \cref{lem:sum_by_parts} to prove \cref{theoAdia}.
\subsection{Properties of operators}
Before giving the detailed lemmas, we provide some properties of the operators from \cite{DKS98}.
First we give properties of the adiabatic operators and the projectors onto the subspaces.
\begin{proposition}
For any integers $T, n,m$ and the corresponding discrete time $s = n/T, s' = m/T$, we have $W_T^{A}(s)$ and $U_T^A(s)$ are unitary, and
\begin{align}\label{eq:projU}
U_T^A(s) P_T(0) &= P_T(s)U_T^A(s), \\
P_T(s+1/T)W_T^A(s) &= W_T^A(s)P_T(s), \\
P_T (s) U_T (s) U_T^\dagger (s') P_T(s') &=
P_T (s) v_{T} (s, s') U_T^A (s)
{U_T^A}^\dagger(s') P_T(s'), \label{eq:DKS_ansatz_P} \\
Q_T (s) U_T (s) U_T^\dagger (s') Q_T(s') &=
Q_T (s) v_{T} (s, s') U_T^A (s)
{U_T^A}^\dagger(s') Q_T(s'), \label{eq:DKS_ansatz_Q} \\
\label{eq:MainEq}
P_T (s + 1/T) W_T (s) P_T (s) &=
P_T (s + 1/T) v_T (s) W_T^A (s) P_T (s),\\
Q_T (s + 1/T) W_T (s) Q_T (s) &=
Q_T (s + 1/T) v_T (s) W_T^A (s) Q_T (s).
\end{align}
\end{proposition}
See Eqs.~(8) to (9) and the accompanying discussion in \cite{DKS98} for explanation of these properties.
Next we consider properties of the \emph{wave operator} $\Omega_T(s)$, the \emph{ripple operator} $\Theta_T (s)$ and the kernel function $K_T(s)$.
One can simply prove that the ripple operator is a rotation of the operator $V_T$, and $\Omega_T$ satisfies a discrete form of the Volterra equation.
The key results are as in the following proposition, which is equivalent to Eqs.~(19) and (20) from \cite{DKS98}.
\begin{proposition}\label{prop:volterra}
For any integer $T,n$ and the discrete time $s = n/T$, we have
\begin{equation}\label{eqn:Theta_V}
\Theta_T (s) =
{U_T^A}^\dagger (s + 1/T) V_T^\dagger (s) {U_T^A} (s + 1/T).
\end{equation}
and the Volterra equation
\begin{equation}\label{eqn:Volterra}
\Omega_T(n/T) = I - \frac{1}{T}\sum_{m=0}^{n-1} K_T(m/T)\Omega_T(m/T).
\end{equation}
\end{proposition}
Finally, we provide the definitions of the $\Tilde{X}$, $A$, $B$ and $Z$ operators.
They are
\begin{align}
\label{eq:Xtilde}
\Tilde{X}(s) & \coloneqq -\frac{1}{2\pi i}\oint_{\Gamma_T(s)}R_T(s,z)X(s)R_T(s,z)dz,\\
\label{eq:A}
A(s) & \coloneqq \left(V_T^{\dagger}\left(s\right)-I\right)W_T^A(s),\\
\label{eq:B}
B(s) & \coloneqq D\Tilde{X}\left(s\right)W_T^A\left(s\right)+DW_T^A\left(s-1/T\right)\Tilde{X}\left(s\right),\\
\label{eq:Z}
Z(s) & \coloneqq T\left(\left[A\left(s\right),\Tilde{X}\left(s\right)\right]+B\left(s\right)\right),
\end{align}
where $R_T(s,z)$ is defined in \cref{eq:resolv}.
At this point the intuition behind these operators is not clear, but we will see later that these operators are related to a summation by parts formula and can simplify the notation.
The definitions of $\Tilde{X}(s)$ and $Z(s)$ are equivalent to Eqs.~(21) and (25) of \cite{DKS98}, and $A(s)$ and $B(s)$ are defined in unnumbered equations in the proof of Theorem 1 of that work.
\subsection{Bounding the operators}
Here we show the bounds for operators of interest with explicit dependence in terms of the gap.
A key part of the method is that we will need to consider a contour $\Gamma_T(s)$ that encloses the spectrum of interest for successive steps of the walk.
In particular, we will use the notation $\Gamma_T(s,k)$ to indicate a contour that encloses the spectrum of interest for $k+1$ successive steps of the walk.
Moreover, for $\Gamma_T(s,k)$ we will take the specific contour that passes in straight lines from the center through the gaps in the spectrum, as shown in \cref{fig:contour_1} and \cref{fig:contour_2}.
Those figures indicate that the contour is closed by an arc at radius 2.
We will take the closure of the contour to be at a distance that approaches infinity for the contours $\Gamma_T(s,k)$.
The results can be obtained by taking the closure at a finite radius then taking that radius to infinity, but for simplicity of the explanation we will not give that limit explicitly except for one illustrative example.
Note that we will only take this limit when the integrand approaches zero more quickly than $1/|z|$.
That will be true for all the contour integrals we consider except that for $P_T(s)$.
\begin{figure}
\caption{Illustration of the choice of the contour $\Gamma_T(s,k)$ for $k = 1$.
That is, there are two successive steps of the walk, and we would need to consider the spectrum for both.
We need to be able to use a contour that separates out the spectrum of interest for both steps of the walk.
This ensures that we have projectors onto the spectrum of interest that are consistent for both steps, with a gap between the contour and the eigenvalues.
We do not allow eigenvalues of interest to cross the gap between one step and the next.
Again we show a contour with radius $2$, but we would take the infinity limit of the radius. }
\label{fig:contour_2}
\end{figure}
We start with the bounds for $DP_T$ and $D^{(2)}P_T$, which can be obtained by direct calculations from the definitions.
\begin{lemma}\label{lem:P}
For any integer $T$ and $n$ and the corresponding discrete time $s=n/T$ we have
\begin{equation}
\label{eq:Dp}
\|DP_T(s)\|\leq \frac{2c_1(s)}{T\Delta_1(s)},
\end{equation}
and
\begin{equation}\label{eqn:D2P}
\|D^{(2)}P_T(s)\| \leq \frac{\mathcal{G}_{T,1}(s)}{T^2},
\end{equation}
with
\begin{equation}\label{eq:GT1}
\mathcal{G}_{T,1}(s) \coloneqq \frac{ c_1(s)^2+c_1(s)c_1(s+1/T)}{\pi (1-\cos(\Delta_2(s)/2))} + \frac{2c_2(s)}{\Delta_2(s)}.
\end{equation}
\end{lemma}
See \cref{ap:proofDP} for the lengthy proof of this result.
Now we move on to bounding the finite difference of the kernel function and the adiabatic walk operator.
The key here is to express and bound the operator $V_T$, because it is related to both the kernel and the adiabatic walk operator.
First, we re-express $V_T$ in terms of $P_T$.
\begin{lemma}\label{lem:V_v2}
For a discrete time $s$, we have
\begin{equation}\label{eqn:def_remainder_V}
V_T(s) = \mathcal{F}_T(s)\left[I + DP_T(s)(2P_T(s)-I)\right],
\end{equation}
where
\begin{equation}
\label{eq:def_F}
\mathcal{F}_T(s) \coloneqq \left[I-\left(DP_T(s)\right)^2 \right]^{-1/2}.
\end{equation}
\end{lemma}
\begin{proof}
By the definition of $V_T(s)$ in \cref{eq:V},
\begin{align}
\label{eq:Vtaprox}
V_T(s)&=\left[I-\left(P_T(s)-P_T\left(s+1/T\right)\right)^2\right]^{-1/2} S_T\left(s+1/T,s\right) \nonumber\\
&=\left[I-\left(P_T(s)-P_T\left(s+1/T\right)\right)^2\right]^{-1/2} \left[I-P_T(s)-P_T\left(s+1/T\right)+2P_T\left(s+1/T\right) P_T(s)\right] \nonumber\\
&=\left[I-\left(P_T(s)-P_T\left(s+1/T\right)\right)^2 \right]^{-1/2} \left[I+\left(P_T\left(s+1/T\right)-P_T(s)\right)\left(2P_T(s)-I\right)\right]\nonumber \\
&=\mathcal{F}_T(s)\left[I + DP_T(s)(2P_T(s)-I)\right].
\end{align}
\end{proof}
That enables us to bound the difference of $V_T$ from the identity.
\begin{lemma}\label{lem:V_bound2}
For a discrete time $s$
\begin{equation}
\|V_T(s) - I\| \leq \|\mathcal{F}_T(s) - I\| + \|DP_T(s)\|\|\mathcal{F}_T(s) \|.
\end{equation}
\end{lemma}
\begin{proof}
From \cref{lem:V_v2} we have
\begin{equation}
V_T(s) - I = \mathcal{F}_T(s) + \mathcal{F}_T(s)(DP_T(s)(2P_T(s)-I)) -I.
\end{equation}
Then the triangle inequality gives
\begin{align}
\|V_T(s) - I\| &\leq \|\mathcal{F}_T(s)-I\| + \|\mathcal{F}_T(s)(DP_T(s)(2P_T(s)-I))\| \nonumber \\
&= \|\mathcal{F}_T(s)-I\| + \|\mathcal{F}_T(s)DP_T(s)\| ,
\end{align}
where in the second line we have used the fact that $2P_T(s)-I$ is a unitary reflection operator.
The inequality $\|\mathcal{F}_T(s)DP_T(s)\| \leq \|\mathcal{F}_T(s)\| \, \|DP_T(s)\|$ then gives the bound required.
\end{proof}
Next we bound the change in $V_T$.
\begin{lemma}\label{lem:DV_v2}
For a discrete time $s$,
\begin{align}
\left\|DV_T(s)\right\| &\leq \left(1+\|DP_T(s+1/T)\|\right)\left\|D^{(2)}P_T(s)\right\| \mathcal{D}_3\left( \max(\|DP_T(s+1/T)\|,\|DP_T(s)\|) \right)\nonumber\\
&\quad+\|\mathcal{F}_T(s)\|\left(\left\|D^{(2)}P_T(s)\right\| + 2 \|DP_T(s)\|^2\right),
\end{align}
with
\begin{equation}
\mathcal{D}_3(z)\coloneqq \frac{z}{(1-z^2)^{3/2}}.
\end{equation}
\end{lemma}
\begin{proof}
For the difference operator $D$ there is the product rule $D(X(s)Y(s))=DX(s)Y(s+1/T)+X(s)DY(s)$ for any two operators $X(s)$ and $Y(s)$. Then, from \cref{lem:V_v2} we have
\begin{align}
D V_T(s) &= D\mathcal{F}_T(s) [I + DP_T(s+1/T)(2P_T(s+1/T)-I)] + \mathcal{F}_T(s)D\left[DP_T(s)(2P_T(s)-I)\right]\nonumber\\
&=D\mathcal{F}_T(s) [I + DP_T(s+1/T)(2P_T(s+1/T)-I)]\nonumber\\ &\quad+ \mathcal{F}_T(s)\left[D^{(2)}P_T(s)(2P_T(s+1/T)-I)+2(DP_T(s))^2\right].
\end{align}
By the triangle inequality and using the fact that the reflection operator is unitary, we get
\begin{equation}\label{eq:l9tmp}
\|D V_T(s)\|\leq\|D\mathcal{F}_T(s)\| \left(1 + \|DP_T(s+1/T)\|\right) + \|\mathcal{F}_T(s)\|\left(\|D^{(2)}P_T(s)\| + 2\|DP_T(s)\|^2\right).
\end{equation}
Now, from the Taylor expansion of $\mathcal{F}_T$
\begin{equation}\label{eqn:F_Taylor}
\mathcal{F}_T(s) = I + \sum_{k=1}^{\infty} \frac{\Pi_{j=1}^{k}(2j-1)}{2^k k!}(DP_T(s))^{2k},
\end{equation}
the bound of $D\mathcal{F}_T$ in terms of $P_T$ can be computed, i.e.
\begin{align}
\| \mathcal{F}_T(s+1/T) - \mathcal{F}_T(s) \| &= \left\| \sum_{k=1}^{\infty} \frac{\Pi_{i=1}^{k}(2i-1)}{2^k k!} \left[ \left(DP_T(s+1/T)\right)^{2k} - \left(DP_T(s)\right)^{2k}\right] \right\| \nonumber \\
&= \left\| \sum_{k=1}^{\infty} \frac{\Pi_{i=1}^{k}(2i-1)}{2^k k!} \sum_{j=0}^{2k-1}\left(DP_T(s+1/T)\right)^{j} \left[ DP_T(s+1/T)-DP_T(s)\right] \left(DP_T(s)\right)^{2k-1-j} \right\| \nonumber \\
&\le \sum_{k=1}^{\infty} \frac{\Pi_{i=1}^{k}(2i-1)}{2^k k!} \left\|D^{(2)}P_T(s)\right\|\sum_{j=0}^{2k-1}\|\left(DP_T(s+1/T)\right)^{j} \left(DP_T(s)\right)^{2k-1-j}\| \nonumber \\
&\le \left\|D^{(2)}P_T(s)\right\|\sum_{k=1}^{\infty} \frac{\Pi_{i=1}^{k}(2i-1)}{2^k k!} (2k) \left[ \max(\|DP_T(s+1/T)\|,\|DP_T(s)\|) \right]^{2k-1} \nonumber \\
&= \left\|D^{(2)}P_T(s)\right\| \mathcal{D}_3\left( \max(\|DP_T(s+1/T)\|,\|DP_T(s)\|) \right),
\end{align}
where we have used the Taylor expansion of the function $\mathcal{D}_3$.
Substituting this into \cref{eq:l9tmp} gives the bound required.
\end{proof}
\begin{lemma}\label{lem:WAv3}
For any discrete time $s$ with $W_T^A(s)$ defined as in \cref{eq:Wa}, we have
\begin{equation}
\label{eqn:DWA}
\|DW_T^A(s)\| \leq \frac{c_1(s)}{T}+ \|DV_T(s)\|.
\end{equation}
\end{lemma}
\begin{proof}
According to the definition of $W_T^A(s)$ as $V_T(s)W_T(s)$,
\begin{align}
DW_T^A(s) = DV_T(s) W_T(s+1/T) + V_T(s) DW_T(s). \end{align}
Since $W_T$ and $V_T$ are unitary, and using the triangle inequality, we have
\begin{equation}
\left\|DW_T^A(s)\right\| \leq \left\|DV_T(s)\right\| + \left\|DW_T(s)\right\|. \nonumber\\
\label{eq:upperDWa}
\end{equation}
Using \cref{eqn:assump1} with $k=1$ for $\left\|DW_T(s)\right\|$ then gives \cref{eqn:DWA}.
\end{proof}
\begin{lemma}\label{lem:DOmega2}
For any discrete time $s$, $\Omega_T$ as defined in \cref{eq:Omegadef}, and $\mathcal{F}_T$ defined as in \cref{eq:def_F}, we have the upper bound on $D\Omega_T(s)$
\begin{equation}
\|D\Omega_T(s)\| \leq \|\mathcal{F}_T(s) - I\| + \|DP_T(s)\|\|\mathcal{F}_T(s) \|.
\end{equation}
\end{lemma}
\begin{proof}
From Proposition~\ref{prop:volterra} and \cref{eqn:Volterra} we see that the difference between $\Omega_T\left(s\right)$ and $\Omega_T\left(s+1/T\right)$ is the term in the sum with $m=n$.
That gives
\begin{align}
D\Omega_T\left(s\right) &= - \frac{1}{T} K_T(s)\Omega_T(s) \nonumber\\
&= (\Theta_T(s)-I)\Omega_T(s) \nonumber\\
&= {U_T^A}(s+1/T)^{\dagger} (V_T(s)-I)^{\dagger}U_T^A(s+1/T)\Omega_T(s).
\end{align}
In the third line we have used \cref{eqn:Theta_V} for $\Theta_T(s)$.
Since $U_T^A$ and $\Omega_T$ are unitary, we have
\begin{equation}
\| D\Omega_T\left(s\right) \| = \|(V_T(s)-I)^{\dagger}\| = \|V_T(s)-I\|.
\end{equation}
Then the desired bound follows from \cref{lem:V_bound2}.
\end{proof}
Finally we summarize the bounds for the operators related to the summation by parts formula.
\begin{lemma}\label{lem:Xt}
For any discrete time $s$ in $\tilde{X}(s)$ as defined in \cref{eq:Xtilde}, and any bounded operator $X(s)$, we have
\begin{equation}\label{eqn:Xt_bound}
\left\|\tilde{X}(s)\right\| \leq \frac{2}{\Delta_0(s)}\|X(s)\|,
\end{equation}
and
\begin{equation}
\left\|D\Tilde{X}(s)\right\| \leq \frac{2}{\Delta_1(s)}\left\|DX(s)\right\| + \frac{2c_1(s)}{\pi T (1-\cos(\Delta_1(s)/2))} \|X(s)\|.
\end{equation}
\end{lemma}
\begin{proof}
The bound for $\tilde{X}$ directly follows from the definition \cref{eq:Xtilde} and choosing an appropriate contour $\Gamma_T(s,0)$.
As shown in \cref{fig:contour_1}, the contour passes in a straight line from the centre through both gaps, and has a circular arc of radius 2 between these two straight lines.
That is, \cref{eq:Xtilde} gives
\begin{align}
\|\Tilde{X}(s)\| & \le \frac{1}{2\pi}\oint_{\Gamma_T(s,0)} \| R_T(s,z)\|^2 \|X(s)\| \, |dz| \nonumber \\
& \le \frac{1}{2\pi} \|X(s)\| \frac{4\pi}{\Delta_0(s)} = \frac{2}{\Delta_0(s)}\|X(s)\|,
\end{align}
where we have used \cref{eq:CountInt1}, but replaced $\Delta_1(s)$ with $\Delta_0(s)$ because we need only consider the eigenvalues for a single step of the walk.
For $D\Tilde{X}(s)$, using $\Gamma_T(s,1)$ (for two consecutive steps of the walk), and using
\cref{eq:Xtilde} we have
\begin{align}
D\Tilde{X}(s)&=-\frac{1}{2\pi i}\oint_{\Gamma_T(s,1)}\left(R_T\left(s+\frac{1}{T},z\right)X\left(s+\frac{1}{T}\right)R_T\left(s+\frac{1}{T},z\right)-R_T\left(s,z\right)X\left(s\right)R_T\left(s,z\right)\right)dz\nonumber\\
&=-\frac{1}{2\pi i}\oint_{\Gamma_T(s,1)}R_T\left(s+\frac{1}{T},z\right)DX\left(s\right)R_T\left(s+\frac{1}{T},z\right)dz\nonumber\\
&\quad -\frac{1}{2\pi i}\left(\oint_{\Gamma_T(s,1)}R_T\left(s,z\right)X\left(s\right)DR_T\left(s,z\right)dz+\oint_{\Gamma_T(s,1)}DR_T\left(s,z\right)X\left(s\right)R_T\left(s+\frac{1}{T},z\right)dz\right).
\end{align}
Using \cref{eq:contProj} we have
\begin{equation}
D R_T(s,z) = - R_T\left(s+\frac{1}{T},z\right)DW_T\left(s\right) R_T\left(s,z\right),
\end{equation}
so
\begin{align}
\left\|DR_T\left(s,z\right)\right\| &\leq
\left\| R_T\left(s+\frac{1}{T},z\right)\right\| \left\|DW_T\left(s\right)\right\| \left\| R_T\left(s,z\right)\right\| \\
& \leq \frac{c_1(s)}{T}\left\|R_T\left(s+\frac{1}{T},z\right)\right\|\left\|R_T\left(s,z\right)\right\| .
\end{align}
We can therefore write an upper bound as
\begin{align}
\|D\Tilde{X}(s)\| &\le \frac{\left\| DX\left(s\right)\right\|}{2\pi} \oint_{\Gamma_T(s,1)}\left\|R_T\left(s+\frac{1}{T},z\right)\right\|^2 |dz| \nonumber\\
&\quad +\frac{\left\| X\left(s\right)\right\|}{2\pi}\left(\oint_{\Gamma_T(s,1)}\left\| R_T\left(s,z\right)\right\|^2 \left\|R_T\left(s+\frac 1T,z\right)\right\| |dz| + \oint_{\Gamma_T(s,1)}\left\| R_T\left(s,z\right)\right\| \left\|R_T\left(s+\frac{1}{T},z\right)\right\|^2 |dz|\right).
\end{align}
Using the bounds on the contour integrals given in \cref{eq:CountInt1} and \cref{eq:CountInt2}, we then get
\begin{align}
\left\|D\Tilde{X}(s)\right\| &\leq \frac{1}{2\pi} \left\|DX(s)\right\|\frac{4\pi}{\Delta_1(s)} + \frac{1}{\pi} \|X(s)\| \frac{c_1(s)}{T} \frac{2}{1-\cos(\Delta_1(s)/2)} \nonumber\\
& = \frac{2}{\Delta_1(s)}\left\|DX(s)\right\| + \frac{2c_1(s)}{\pi T (1-\cos(\Delta_1(s)/2))} \|X(s)\|.
\end{align}
\end{proof}
\begin{lemma}\label{lem:ABZ2_v2}
For a discrete time $s$ in $A(s)$, $B(s)$ and $Z(s)$ defined in \cref{eq:A,eq:B,eq:Z} respectively, and any bounded operator $X(s)$, we have
\begin{equation}
\label{eq:boundA}
\|A(s)\|\leq \|\mathcal{F}_T(s) - I\| + \|DP_T(s)\|\|\mathcal{F}_T(s) \|,
\end{equation}
\begin{align}
\label{eq:BoundB}
\left\|B\left(s\right)\right\| &\leq \frac{2}{\Delta_1(s)}\left\|DX(s)\right\| + \frac{2c_1(s)}{\pi T (1-\cos(\Delta_1(s)/2))} \|X(s)\|\nonumber\\
&\quad+ \frac{2}{\Delta_0(s)}\left (\frac{c_1(s-1/T)}{T}+ \|DV_T(s-1/T)\|\right)\|X(s)\|,
\end{align}
and
\begin{align}\label{eq:Zbound}
\|Z(s)\| &\leq \frac{4T}{\Delta_0(s)} \left(\|\mathcal{F}_T(s) - I\| + \|DP_T(s)\|\|\mathcal{F}_T(s) \|\right)\|X(s)\| + \frac{2T}{\Delta_1(s)}\left\|DX(s)\right\| \nonumber\\
&\quad+ \frac{2c_1(s)}{\pi (1-\cos(\Delta_1(s)/2))} \|X(s)\| + \frac{2}{\Delta_0(s)}\left (c_1(s-1/T)+ T\|DV_T(s-1/T)\|\right)\|X(s)\|.
\end{align}
\end{lemma}
\begin{proof}
From the definition of $A$ in \cref{eq:A}, we have
\begin{equation}
\|A(s)\|=\left\|\left(V_T\left(s\right)^{\dagger}-I\right)W_T^A(s) \right\|\leq \left\|\left(V_T\left(s\right)^{\dagger}-I\right) \right\| \left\|W_T^A(s) \right\|
= \left\|\left(V_T\left(s\right)^{\dagger}-I\right) \right\|.
\end{equation}
The bound for $\|A\|$ follows from \cref{lem:V_bound2}.
For $B$, from \cref{eq:B}, we have that
\begin{equation}
\left\|B\left(s\right)\right\| \leq \left\| D\Tilde{X}\left(s\right)W_T^A\left(s\right)\right\|+\left\| DW_T^A\left(s-1/T\right)\Tilde{X}\left(s\right)\right\|
\leq \left\| D\Tilde{X}\left(s\right)\right\|+\left\| DW_T^A\left(s-1/T\right)\right\| \left\|\Tilde{X}\left(s\right)\right\|.
\end{equation}
By inserting the bounds previously computed for $D\tilde{X}$ (in \cref{lem:Xt}) and $DW_T^A$ (in \cref{lem:WAv3}), the desired bound for $B$ is established.
Finally, for $Z$ the definition \cref{eq:Z} immediately gives
\begin{equation}
\left\|Z\left(s\right)\right\|\leq T\left(2\left\|A\left(s\right)\right\| \left\|\tilde{X}\left(s\right)\right\| + \left\|B\left(s\right)\right\|\right).
\end{equation}
The bounds previously computed in \cref{eq:BoundB,eqn:Xt_bound,eq:boundA} then give the upper bound required.
\end{proof}
The summation by parts formula, which is presented as our next lemma here, is given in Theorem 1 of \cite{DKS98} with a typo in the sign of both operators $\mathcal{S}$ and $\mathcal{B}$.
Here we correct the sign slightly differently for the two quantities, taking $\mathcal{B}$ to be the negative of the $\mathcal{B}$ defined in \cite{DKS98}, and taking $\mathcal{S}$ to be the same but placing a minus sign in the statement of the theorem (so there is $\mathcal{B}-\mathcal{S}/T$).
Throughout the lemma and its proof we will encounter slight shift of the discrete time very frequently. To simplify the notation, for any positive integer $n$, we define $n_+ = n+1$ and $n_- = n-1$.
\begin{lemma}[Summation by parts formula]
\label{lem:sum_by_parts}
Let $W_T(s)$, $s\in \mathbb{Z}/T$, be a sequence of unitaries,
and suppose that $X(s)$ and $Y(s)$ are sequences of operators.
Then
\begin{equation}
\sum_{n=1}^{l}Q_0 U^{A\dagger}_T\left(\frac{n}{T}\right )X\left(\frac{n}{T}\right) U^{A}_T\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right) = \mathcal{B} - \frac{1}{T}\mathcal{S},
\end{equation}
where $P_0=P_T(0)$ and $Q_0=Q_T(0)$,
\begin{equation}
\label{eq:Bou}
\mathcal{B} = Q_0U^{A\dagger}_T\left(\frac{l}{T}\right)\Tilde{X}\left(\frac{l_+}{T}\right)U^{A}_T\left(\frac{l_+}{T}\right)P_0Y\left(\frac{l_+}{T}\right) - Q_0U_T^{A\dagger}(0)\Tilde{X}\left(\frac{1}{T}\right)U^{A}_T\left(\frac{1}{T}\right)P_0Y\left(\frac{1}{T}\right),
\end{equation}
is the boundary term and
\begin{equation}
\label{eq:Sum}
\mathcal{S}=\sum_{n=1}^{l} Q_0 U^{A\dagger}_T\left(\frac{n}{T}\right) \left(Z\left(\frac{n}{T}\right) U_T^{A}\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right) + \Tilde{X}\left(\frac{n_+}{T}\right) W^A_T\left(\frac{n}{T}\right) U_T^A\left(\frac{n}{T}\right) P_0 T DY\left(\frac{n}{T}\right)\right),
\end{equation}
is the sum.
\end{lemma}
As we are making a correction to the theorem, and it is quite lengthy, we give a proof in \cref{ap:sum_parts}.
\subsection{The first discrete adiabatic theorem}
\label{sec:dat1}
We now give the complete explicit form of the discrete adiabatic theorem.
\begin{theorem}[The First Discrete Adiabatic Theorem]
\label{theoAdia}
Let $U_T(s) = \prod_{l = 0}^{sT-1} W_T\left(l/T\right)$ for $s\in \mathbb{Z}/T$ be a product of unitary operators $W_T\left(l/T\right)$ as per
\cref{eq:U_as_product_of_W}, and let $U_T^A(s)$ be the corresponding ideal adiabatic evolution that maps an eigenstate of $W_T(0)$ to the corresponding eigenstate of $W_T(s)$.
Suppose further that the operators $W_T (s)$ satisfy
$\left\|D^{(k)}W_T(s)\right\|\leq c_k(s)/T^k$ for $k = 1,2$, as per \cref{def:difs},
we consider the gaps $\Delta_k(s)$ as defined in \cref{def:gaps},
and $T \geq \max_{s\in [0,1]} (2c_1(s)/\Delta_1(s))$.
Then for any time $s$, we have
\begin{align}
\label{eq:mainEqTheo}
& \quad \left\|U_T (s) - U_T^{A}(s) \right\| \nonumber \\
& \leq\frac{4}{\Delta_0(1/T)}\mathcal{D}_2\left(\frac{2c_1(0)}{T\Delta_1(0)}\right) + \frac{4}{\Delta_0(s)}\mathcal{D}_2\left(\frac{2c_1(s-1/T)}{T\Delta_1(s-1/T)}\right) + 2\mathcal{D}_2\left(\frac{2c_1(s-1/T)}{T\Delta_1
(s-1/T)}\right) \nonumber\\
& \quad + \sum_{n=1}^{sT-1}4\left(\frac{1}{\Delta_0(n_+/T)} + \frac{2}{\Delta_0(n/T)}\right)\mathcal{D}_2\left(\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right)\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right) \nonumber\\
&\quad + \sum_{n=1}^{sT-1}\frac{4\mathcal{G}_{T,3}(n_-/T)}{T^2\Delta_1(n/T)} + \sum_{n=1}^{sT-1} \frac{4c_1(n/T)}{\pi T (1-\cos(\Delta_1(n/T)/2))} \mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1
(n_-/T)}\right)\nonumber\\
&\quad+ \sum_{n=1}^{sT-1}\frac{4 \mathcal{G}_{T,4}(n_-/T)}{T\Delta_0(n/T)}\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1
(n_-/T)}\right) + \sum_{n=0}^{sT-1} \frac{24c_1(n/T)^2}{T^2\Delta_1(n/T)^2} + \sum_{n=0}^{sT-1}\frac{4c_1(n/T)^2}{T^2\Delta_1(n/T)^2}\left(1-\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right)^{-1}, \end{align}
where
\begin{align}
\label{eq:D_i}
\mathcal{D}_1(z) &\coloneqq \frac{1}{\sqrt{1-z^2}}, \quad \mathcal{D}_2(z) \coloneqq \sqrt{\frac{1+z}{1-z}} - 1, \quad \mathcal{D}_3(z)\coloneqq \frac{z}{(1-z^2)^{3/2}},\\
\label{eq:G1}
\mathcal{G}_{T,1}(s) &\coloneqq \frac{ c_1(s)^2+c_1(s)c_1(s+1/T)}{\pi (1-\cos(\Delta_2(s)/2))} + \frac{2c_2(s)}{\Delta_2(s)}, \\
\label{eq:G2}
\mathcal{G}_{T,2}(s) &\coloneqq \mathcal{G}_{T,1}(s)\mathcal{D}_3\left( \max\left(\frac{2c_1(s+1/T)}{T\Delta_1(s+1/T)},\frac{2c_1(s)}{T\Delta_1(s)}\right) \right), \\
\label{eq:G3}
\mathcal{G}_{T,3}(s) &\coloneqq \mathcal{G}_{T,2}(s)\left(1 + \frac{2c_1(s)}{T\Delta_1(s)}\right) + \mathcal{D}_1\left(\frac{2c_1(s)}{T\Delta_1(s)}\right)\left(\mathcal{G}_{T,1}(s) + \frac{8c_1(s)^2}{\Delta_1(s)^2}\right), \\
\label{eq:G4}
\mathcal{G}_{T,4}(s) &\coloneqq \frac{\mathcal{G}_{T,3}(s)}{T} + c_1(s).
\end{align}
\end{theorem}
\begin{proof}
Starting from the definition of $K_T$ and Proposition~\ref{prop:volterra}, for any discrete time $s$,
\begin{align}
\|U_T(s)-U_T^A(s)\| &= \|\Omega_T(s)-I\| \nonumber\\
&= \left\|\frac{1}{T}\sum_{n=0}^{sT-1}K_T\left(\frac{n}{T}\right)\Omega_T\left(\frac{n}{T}\right)\right\| \nonumber\\
&= \left\|\sum_{n=0}^{sT-1}\left(I - \Theta_T\left(\frac{n}{T}\right)\right)\Omega_T\left(\frac{n}{T}\right)\right\| \nonumber\\
&= \left\|\sum_{n=1}^{sT}{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)\Omega_T\left(\frac{n_-}{T}\right)\right\|.
\end{align}
Note that in the summation by parts formula only the ``off-diagonal'' term is considered.
This motivates us to further split the sum into ``diagonal'' and ``off-diagonal'' terms as
\begin{align}
\|U_T(s)-U_T^A(s)\|&= \left\|\sum_{n=1}^{sT}{(P_0+Q_0)U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)(P_0+Q_0)\Omega_T\left(\frac{n_-}{T}\right)\right\| \label{eqn:main_bound_split}\\ &\leq \left\|\sum_{n=1}^{sT}P_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)P_0\Omega_T\left(\frac{n_-}{T}\right)\right\| \label{eq:diag1}\\
& \quad + \left\|\sum_{n=1}^{sT}Q_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)Q_0\Omega_T\left(\frac{n_-}{T}\right)\right\| \label{eq:diag2}\\
& \quad + \left\|\sum_{n=1}^{sT}Q_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)P_0\Omega_T\left(\frac{n_-}{T}\right)\right\| \label{eq:offdiag1}\\
& \quad + \left\|\sum_{n=1}^{sT}P_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)Q_0\Omega_T\left(\frac{n_-}{T}\right)\right\|\label{eq:offdiag2},
\end{align}
where \cref{eq:diag1,eq:diag2} are the diagonal and \cref{eq:offdiag1,eq:offdiag2} are the off-diagonal components. For the ``diagonal'' term, it is possible to show
\begin{align}
&\quad \left\|\sum_{n=1}^{sT}P_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)P_0\Omega_T\left(\frac{n_-}{T}\right)\right\| \nonumber\\
&\leq \sum_{n=0}^{sT-1} \left\|I-\mathcal{F}_T\left(\frac{n}{T}\right)\right\|\left(1+\left\|DP_T\left(\frac{n}{T}\right)\right\|\right) + 3\sum_{n=0}^{sT-1}\left\|DP_T\left(\frac{n}{T}\right)\right\|^2.
\end{align}
This is shown in \cref{ap:sec_diag}, where the result is given in \cref{eq:diag_p2}.
The reasoning for the term with $Q_0$ is identical and gives the same result.
Using \cref{lem:V_v2} one can show
\begin{equation}
\|\mathcal{F}_T(s) - I\|\leq\mathcal{D}_1\left(\frac{2c_1(s)}{T\Delta_1(s)}\right)-1 .
\end{equation}
The steps for deriving the above bound are given in \cref{eq:bounF-i} and the function $\mathcal{D}_1$ is defined in \cref{eq:D_i}. Therefore, one obtains the following bound for the ``diagonal'' term
\begin{align}
& \quad \left\|\sum_{n=1}^{sT}P_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)P_0\Omega_T\left(\frac{n_-}{T}\right)\right\| \nonumber\\
&\leq \sum_{n=0}^{sT-1} \frac{12c_1(n/T)^2}{T^2\Delta_1(n/T)^2} + \sum_{n=0}^{sT-1}\left(\mathcal{D}_1\left(\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right)-1\right)\left(1+\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right) \nonumber\\
& \leq \sum_{n=0}^{sT-1} \frac{12c_1(n/T)^2}{T^2\Delta_1(n/T)^2} + \sum_{n=0}^{sT-1}\frac{2c_1(n/T)^2}{T^2\Delta_1(n/T)^2}\left(1-\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right)^{-1}, \label{eqn:diagonal}
\end{align}
where in the last line we use the inequality $[(1-z^2)^{-1/2}-1](1+z)\le z^2/[2(1-z)]$ for all $0 \leq z < 1 $.
The exact same bound holds for the second diagonal term with $Q_0$ in \cref{eq:diag2}.
The reason is that we have treated the eigenspace of interest and the complementary eigenspace completely symmetrically.
Therefore exactly the same bounds hold with $P_T$ replaced with $Q_T$, and the above bound must continue to hold.
For the ``off-diagonal'' term we can similarly consider only the term with $Q_0$ on the left and $P_0$ on the right as in \cref{eq:offdiag1}, and exactly the same bound will hold for the other off-diagonal term in \cref{eq:offdiag2}.
Using \cref{lem:sum_by_parts} with $X(s) = T(1-V_T^{\dagger}(s-1/T))$ and $Y(s) = \Omega_T(s-1/T)$ (note the slight shift in time), it is possible to show
\begin{align}
& \quad \left\|\sum_{n=1}^{sT}Q_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)P_0\Omega_T\left(\frac{n_-}{T}\right)\right\|\nonumber\\
& \leq \frac{1}{T}\left\|\tilde{X}\left(\frac{1}{T}\right)\right\| + \frac{1}{T}\left\|\tilde{X}\left(s\right)\right\| + \frac{1}{T}\left\|X\left(s\right)\right\|
+ \frac{1}{T^2}\sum_{n=1}^{sT-1} \left\|Z\left(\frac{n}{T}\right)\right\| + \frac{1}{T}\sum_{n=1}^{sT-1} \left\|\tilde{X}\left(\frac{n_+}{T}\right)\right\| \left\|DY\left(\frac{n}{T}\right)\right\|.
\end{align}
See \cref{ap:offD} for the derivation and the result in \cref{eq:offDiag_B}.
By using \cref{lem:P,lem:DOmega2,lem:V_bound2,lem:DV_v2,lem:Xt,lem:ABZ2_v2}, we can show
\begin{align}
\left\|X\left(\frac{n}{T}\right)\right\| &\leq T\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1
(n_-/T)}\right), \\
\left\|\tilde{X}\left(\frac{n}{T}\right)\right\| &\leq \frac{2T}{\Delta_0(n/T)}\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right), \\
\left\|Z\left(\frac{n}{T}\right)\right\| &\leq \frac{4T^2}{\Delta_0\left(n/T\right)} \mathcal{D}_2\left(\frac{2c_1(n/T)}{\Delta_1(n/T)}\right)\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{\Delta_1(n_-/T)}\right) + \frac{2\mathcal{G}_{T,3}(n_-/T)}{\Delta_1(n/T)} \nonumber\\
&\quad+ \frac{2Tc_1(n/T)}{\pi (1-\cos(\Delta_1(n/T)/2))} \mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right) + \frac{2T\mathcal{G}_{T,4}(n_-/T)}{\Delta_0(n/T)}\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right), \\
\left\|DY\left(\frac{n}{T}\right)\right\| &\leq \mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right),
\end{align}
with $\mathcal{D}_2(x)$, $\mathcal{G}_{T,3}(n/T)$ and $\mathcal{G}_{T,4}(n/T)$ given in \cref{eq:D_i,eq:G3,eq:G4} respectively. The bounds of these operators are derived in \cref{ap:offD}; see \cref{eq:Xbnd,eq:Xtildebnd,eq:Zbnd,eq:Ybnd}.
Therefore
\begin{align}
& \quad \left\|\sum_{n=1}^{sT}Q_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)P_0\Omega_T\left(\frac{n_-}{T}\right)\right\| \nonumber\\
& \leq \frac{2}{\Delta_0(1/T)}\mathcal{D}_2\left(\frac{2c_1(0)}{T\Delta_1(0)}\right) + \frac{2}{\Delta_0(s)}\mathcal{D}_2\left(\frac{2c_1(s-1/T)}{T\Delta_1(s-1/T)}\right) + \mathcal{D}_2\left(\frac{2c_1(s-1/T)}{T\Delta_1
(s-1/T)}\right) \nonumber\\
& \quad + \sum_{n=1}^{sT-1}2\left(\frac{1}{\Delta_0(n_+/T)} + \frac{2}{\Delta_0(n/T)}\right)\mathcal{D}_2\left(\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right)\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right) \nonumber\\
&\quad + \sum_{n=1}^{sT-1}\frac{2\mathcal{G}_{T,3}(n_-/T)}{T^2\Delta_1(n/T)} + \sum_{n=1}^{sT-1} \frac{2c_1(n/T)}{\pi T (1-\cos(\Delta_1(n/T)/2))} \mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1
(n_-/T)}\right)\nonumber\\
&\quad+ \sum_{n=1}^{sT-1}\frac{2 \mathcal{G}_{T,4}(n_-/T)}{T\Delta_0(n/T)}\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1
(n_-/T)}\right). \label{eqn:offdiagonal}
\end{align}
Finally, by using \cref{eqn:diagonal} and \cref{eqn:offdiagonal} in \cref{eqn:main_bound_split}, we obtain the required overall bound in \cref{eq:mainEqTheo}.
\end{proof}
\subsection{Proof of the second adiabatic theorem}
Because the first form of the discrete adiabatic theorem is quite complicated, we give a simplified but looser form in \cref{cor:adia}.
In this subsection we provide the proof of that result.
\begin{proof}
The key ideas to obtain \cref{cor:adia} from \cref{theoAdia} are: replace the functions $c_1(s)$ and $c_2(s)$ by \cref{eq:chat}, which take into account neighbouring steps; replace the gaps $\Delta_k(s)$ by $\check{\Delta}(s)$ as defined in \cref{eq:fhat}, which takes into account the minimum gap in neighbouring steps; and bounding the higher-order terms by lower-order terms with a slightly more strict assumption on $T$, that it is no less than $\max_s (4\hat{c}_1(s)/\check{\Delta}(s))$.
We first bound the functions $\mathcal{D}_k$ with simpler expressions.
Recall that the definitions of $\mathcal{D}_k$ are
\begin{equation}
\mathcal{D}_1(z) = \frac{1}{\sqrt{1-z^2}}, \qquad \mathcal{D}_2(z) = \sqrt{\frac{1+z}{1-z}} - 1, \qquad \mathcal{D}_3(z)= \frac{z}{(1-z^2)^{3/2}}.
\end{equation}
Notice that in \cref{theoAdia} all the arguments in $\mathcal{D}_k$ are in the form of $2c_1/(T\Delta_1)$, then, under the assumption on $T$, we are only interested in the case $0 \leq z \leq 1/2$.
Then we have
\begin{align}
\label{eq:upperB_D}
\mathcal{D}_1(z) \leq \xi_1, \qquad
\mathcal{D}_2(z) \leq \xi_2 z, \qquad
\mathcal{D}_3(z) \leq \xi_3 z
\end{align}
with constants $\xi_1 = 2/\sqrt{3}, \xi_2 = 2\sqrt{3}-2, \xi_3 = 8/(3\sqrt{3})$.
Now we move on to the functions $\mathcal{G}_{T,k}$.
For any positive integer $n$, from \cref{theoAdia} we need to bound $\mathcal{G}_{T,3}(n_-/T)$ and $\mathcal{G}_{T,4}(n_-/T)$, which in turn depend on $\mathcal{G}_{T,1}(n_-/T)$ and $\mathcal{G}_{T,2}(n_-/T)$.
Using the inequality $1-\cos(\theta/2) = 2\sin^2(\theta/4) \geq \theta^2/\pi^2$ for all $0\leq \theta \leq \pi$, it is possible to show that
\begin{equation}\label{eq:GT1body}
\mathcal{G}_{T,1}(n_-/T) \leq \frac{2\pi\hat{c}_1(n/T)^2}{\check{\Delta}(n/T)^2} + \frac{2\hat{c}_2(n/T)}{\check{\Delta}(n/T)}.
\end{equation}
That leads to the following upper bounds for the other main functions,
\begin{equation}\label{eq:GT2body}
\mathcal{G}_{T,2}(n_-/T) \leq \frac{4\pi\xi_3 \hat{c}_1(n/T)^3}{T\check{\Delta}(n/T)^3} + \frac{4\xi_3 \hat{c}_1(n/T)\hat{c}_2(n/T)}{T\check{\Delta}(n/T)^2},
\end{equation}
\begin{equation}\label{eq:GT3body}
\mathcal{G}_{T,3}(n_-/T) \leq \left(3\pi\xi_3/2 + (2\pi+8)\xi_1\right)\frac{\hat{c}_1(n/T)^2}{\check{\Delta}(n/T)^2} + \left(3\xi_3/2+2\xi_1\right)\frac{\hat{c}_2(n/T)}{\check{\Delta}(n/T)},
\end{equation}
and
\begin{align}\label{eq:GT4body}
\mathcal{G}_{T,4}(n_-/T) \leq \left(3\pi\xi_3/2 + (2\pi+8)\xi_1\right)\frac{\hat{c}_1(n/T)^2}{T\check{\Delta}(n/T)^2} + \left(3\xi_3/2+2\xi_1\right)\frac{\hat{c}_2(n/T)}{T\check{\Delta}(n/T)} + \hat{c}_1(n/T).
\end{align}
See \cref{sec:uppMainF} for the details, where these bounds are given in \cref{eq:GT1app,eq:GT2app,eq:GT3app,eq:GT4app}.
Plugging all these estimates back to \cref{theoAdia} and using $1-\cos(\theta/2) = 2\sin^2(\theta/4) \geq \theta^2/\pi^2$ again, it is possible to show that
\begin{align}\label{eq:Uerrbndbody}
\|U_T(s) - U_T^A(s)\|
& \leq \frac{8\xi_2\hat{c}_1(0)}{T\check{\Delta}(0)^2}+ \frac{8\xi_2\hat{c}_1(s)}{T\check{\Delta}(s)^2} + \frac{4\xi_2\hat{c}_1(s)}{T\check{\Delta}(s)} + \sum_{n=1}^{sT-1} \left(48\xi_2^2 + 6\pi \xi_3 + (8\pi+32)\xi_1 + 8\pi \xi_2\right) \frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} \nonumber \\
& \quad + \sum_{n=1}^{sT-1}(6\xi_3+8\xi_1)\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2}+ \sum_{n=0}^{sT-1} \frac{(32+8\xi_2) \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^2} \nonumber \\
& \quad + \sum_{n=1}^{sT-1}\left(12\pi\xi_2\xi_3+(16\pi+64)\xi_1\xi_2\right) \frac{\hat{c}_1(n/T)^3}{T^3\check{\Delta}(n/T)^4} + \sum_{n=1}^{sT-1} \left(12\xi_2\xi_3+16\xi_1\xi_2\right)\frac{\hat{c}_1(n/T)\hat{c}_2(n/T)}{T^3\check{\Delta}(n/T)^3} .
\end{align}
This result is derived in \cref{sec:uppMainF}, \cref{eq:Uerrbndapp}.
Finally, for a clear representation in terms of the gap, we slightly modify some terms with $T^3$ on the denominator to $T^2$ by using the bounds $\hat{c}_1(s)/(T\check{\Delta}(s)) \leq 1/4$, then
\begin{align}
\|U_T(s) - U_T^A(s)\|
& \leq \frac{8\xi_2\hat{c}_1(0)}{T\check{\Delta}(0)^2}+ \frac{8\xi_2\hat{c}_1(s)}{T\check{\Delta}(s)^2} + \frac{4\xi_2\hat{c}_1(s)}{T\check{\Delta}(s)} + \sum_{n=1}^{sT-1} \left(48\xi_2^2 + 6\pi \xi_3 + (8\pi+32)\xi_1 + 8\pi \xi_2\right) \frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} \nonumber \\
& \quad + \sum_{n=1}^{sT-1}(6\xi_3+8\xi_1)\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2}+ \sum_{n=0}^{sT-1} \frac{(32+8\xi_2) \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^2} \nonumber \\
& \quad + \sum_{n=1}^{sT-1}\left(3\pi\xi_2\xi_3+(4\pi+16)\xi_1\xi_2\right) \frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} + \sum_{n=1}^{sT-1} \left(3\xi_2\xi_3+4\xi_1\xi_2\right)\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2} \nonumber \\
& = \frac{8\xi_2\hat{c}_1(0)}{T\check{\Delta}(0)^2}+ \frac{8\xi_2\hat{c}_1(s)}{T\check{\Delta}(s)^2} + \frac{4\xi_2 \hat{c}_1(s)}{T\check{\Delta}(s)} \nonumber \\
& \quad + \left(48\xi_2^2 + 6\pi \xi_3 + (8\pi+32)\xi_1 + 8\pi \xi_2 + 3\pi\xi_2\xi_3+(4\pi+16)\xi_1\xi_2\right)\sum_{n=1}^{sT-1}\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} \nonumber \\
& \quad + \left(32+8\xi_2\right)\sum_{n=0}^{sT-1} \frac{ \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^2} + (6\xi_3+8\xi_1+3\xi_2\xi_3+4\xi_1\xi_2)\sum_{n=1}^{sT-1}\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2} \nonumber \\
& \leq \frac{12\hat{c}_1(0)}{T\check{\Delta}(0)^2}+ \frac{12\hat{c}_1(s)}{T\check{\Delta}(s)^2} + \frac{6 \hat{c}_1(s)}{T\check{\Delta}(s)} \nonumber \\
& \quad + 305\sum_{n=1}^{sT-1}\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} + 44\sum_{n=0}^{sT-1} \frac{ \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^2} + 32\sum_{n=1}^{sT-1}\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2},
\end{align}
where the last inequality is derived by plugging the concrete values of $\xi_k$ into the bound and rounding the resulting constants to the closest integers greater than or equal to them.
\end{proof}
\section{Application: solving linear systems}
\label{sec:linsys}
\subsection{Preparing the walker}
In this section we apply \cref{theoAdia}, about adiabatic evolution in the discrete setting, to solve the quantum linear system problem.
In adiabatic quantum computation, one usually uses a Hamiltonian that is a combination of two Hamiltonians as
\begin{equation}
\label{eq:Ham_prob}
H(s) = (1-f(s))H_0 + f(s)H_1,
\end{equation}
where the function $f(s): [0,1] \rightarrow [0,1]$ is called the schedule function.
Normally $H_0$ is the Hamiltonian where the ground state is easy to prepare, and $H_1$ is the one where the ground state encodes the solution of the problem that we are trying to determine.
For the case of linear systems solvers, the ground state of $H(1)$ should encode the normalized solution for a linear system. In other words, for $A\in \mathbb{C}^{N\times N}$ an invertible matrix with $\|A\|=1$, and a normalized vector $\ket{b}\in \mathbb{C}^N$ the goal is to prepare a normalized state $\ket{\tilde{x}}$, which is an approximation of $\ket{x}=A^{-1}\ket{b}/\|A^{-1}\ket{b}\|$.
For precision $\epsilon$ of the approximation, we require $\|\ket{\tilde{x}} - \ket{x}\|\leq \epsilon$.
One can also bound the error in terms of $\|\ket{\tilde{x}}\bra{\tilde{x}} - \ket{x}\bra{x}\|$ as was done in some prior work \cite{PhysRevLett.122.060504,an2019quantum}, which is asymptotically equal (for small error).
Translating this problem to our theorem for the adiabatic evolution, $\ket{\tilde{x}}$ would be the state achieved from the steps of the walk, and $\ket{x}$ would be obtained from the ideal adiabatic evolution.
Beginning with the simplest case, where $A$ is Hermitian and positive definite, one takes the Hamiltonians \cite{an2019quantum}
\begin{equation}
\label{eq:H0}
H_0\coloneqq\begin{pmatrix} 0 & Q_b\\
Q_b & 0
\end{pmatrix},
\end{equation}
and
\begin{equation}
\label{eq:H1}
H_1\coloneqq \begin{pmatrix} 0 & AQ_b\\
Q_bA & 0
\end{pmatrix},
\end{equation}
where $Q_b=I_N-\ket{b}\bra{b}$.
The state $\ket{0,b}$ is an eigenstate of $H_0$ with eigenvalue 0, and one would aim for this to evolve adiabatically to eigenstate $\ket{0,A^{-1}b}$ of $H_1$.
There is also eigenstate $\ket{1,b}$ or both $H_0$ and $H_1$ with the same eigenvalue 0, but it is orthogonal and we will show that there is no crossover in the ideal adiabatic evolution using the walk.
Denoting the condition number of the matrix as $\kappa$, a lower bound for the gap of $H(s)$ is \cite{an2019quantum}
\begin{equation}
\label{eq:gapSc}
\Delta_0(s)= 1- f(s) + f(s)/\kappa.
\end{equation}
Note that according to \cref{def:gaps}, $\Delta_0(s)$ is a lower bound on the exact gap between the eigenvalues, so we use an equality here rather than an inequality.
Since the goal is to get a schedule function which slows down the evolution as the gap becomes small, a standard condition for the schedule is \cite{jansen2007bounds}
\begin{equation}
\label{eq:gapCon}
\dot{f}(s) = d_p\Delta_0^p(s),
\end{equation}
where $f(0)=0$, $p>0$ and $d_p=\int_0^1\Delta_0^{-p}(u)\, du$ is a normalization constant chosen so that $f(1)=1$. It is possible to show that \cite{an2019quantum}
\begin{equation}
\label{eq:sched1}
f(s) = \frac{\kappa}{\kappa - 1}\left[1-\left(1+s\left(\kappa^{p-1}-1\right)\right)^{\frac{1}{1-p}}\right],
\end{equation}
satisfies \cref{eq:gapCon}, but with $\Delta_0(s)$ replaced with the lower bound on the gap from \cref{eq:gapSc}. This schedule function has two properties that have useful applications to estimate the upper bounds for the difference between consecutive walker operators, namely that $f(s)$ is monotonic increasing and that $\dot{f}(s)$ is monotonic decreasing.
Distinct from the continuous version of the adiabatic theorem, in our discrete version of the theorem, we have to take into account the gap between the different groups of eigenvalues of $W_T(s)$ for $s,s+1/T$ and $s+2/T$, as described in \cref{eqn:assump2}.
From the property that the gap function is monotonically increasing we have
\begin{equation}
\label{eq:Gaps}
\Delta_k(s) = 1- f(s+k/T) + f(s+k/T)/\kappa, \quad k=0,1,2.
\end{equation}
In the case where $A$ is not positive definite but Hermitian, the Hamiltonians $H_0$ and $H_1$ can be modified.
In \cite{PhysRevLett.122.060504} the authors show that one can use a larger Hilbert space for both $H_0$ and $H_1$ as
\begin{equation}
H_0=\sigma_+ \otimes \left[(\sigma_z \otimes I_N)Q_{+,b}\right] + \sigma_- \otimes \left[Q_{+,b}(\sigma_z \otimes I_N)\right],
\end{equation}
where $Q_{+,b} = I_{2N}-\ket{+,b}\bra{+,b}$ and $\ket{\pm}=\frac{1}{\sqrt{2}}(\ket{0}\pm\ket{1})$, and
\begin{equation}
\label{eq:H1_nonP}
H_1=\sigma_+ \otimes \left[(\sigma_x \otimes A)Q_{+,b}\right] + \sigma_- \otimes \left[Q_{+,b}(\sigma_x \otimes A)\right].
\end{equation}
Here two ancilla qubits are needed to enlarge the matrix block.
The solution of the linear system problem can be obtained if we can prepare the zero-energy state $\ket{0,+,b}$ of $H_0$.
By replacing the Hamiltonians in \cref{eq:Ham_prob} for solving the QLSP it is possible to show that the spectral gap of $H(s)$ is lower bounded as $\Delta_0(s) \geq \sqrt{(1-f(s))^2 + (f(s)/\kappa)^2}$ \cite{PhysRevLett.122.060504}.
To avoid the need to use this formula, you can use the relation that for $0\leq f(s) \leq 1$,
\begin{equation}
\sqrt{(1-f(s))^2 + (f(s)/\kappa)^2} \geq (1-f(s) + f(s)/\kappa)/\sqrt{2}.
\end{equation}
When $A$ is not positive definite, you can keep using the same schedule function from \cref{eq:sched1}, but the spectral gaps can instead be lower bounded by
\begin{equation}
\label{eq:GapsA_non}
\Delta'_k(s) = \left(1- f(s+k/T) + f(s+k/T)/\kappa\right)/\sqrt{2}, \quad k=0,1,2.
\end{equation}
The standard approach when $A$ is non-Hermitian is to construct a Hermitian matrix as
\begin{equation}\label{eq:Avec}
\mathbf{A}\coloneqq\begin{pmatrix} 0 & A\\
A^{\dagger} & 0
\end{pmatrix},
\end{equation}
and use
\begin{equation}\label{eq:bvec}
\mathbf{b}\coloneqq\begin{pmatrix} 0\\
b
\end{pmatrix}.
\end{equation}
In our case we adopt a slightly different approach, and instead of replacing $A$ in \cref{eq:H1}, we replace $\sigma_x \otimes A$ in \cref{eq:H1} by $\mathbf{A}$.
As we will show, this gives the same lower bound on the gap without further expanding the dimension.
Then you have the final Hamiltonian
\begin{equation}
H_1=\sigma_+ \otimes \left[\mathbf{A} Q_{\mathbf{b}}\right] + \sigma_- \otimes \left[Q_{\mathbf{b}} \mathbf{A}\right].
\end{equation}
Now define
\begin{equation}\label{eq:Af}
A(f) \coloneqq (1-f) \sigma_z \otimes I_N + f \mathbf{A} =
\begin{pmatrix} (1-f)I & f A\\
f A^{\dagger} & -(1-f)I
\end{pmatrix}
\end{equation}
so
\begin{equation}\label{eq:Hsencoding}
H(s) = (1-f(s)) H_0 + f(s) H_1 = \begin{pmatrix}
0 & A(f(s)) Q_{\mathbf{b}} \\
Q_{\mathbf{b}} A(f(s)) & 0
\end{pmatrix}
\end{equation}
Then it is found that
\begin{equation}
H^2(s) = \begin{pmatrix}
A(f(s)) Q_{\mathbf{b}} A(f(s)) & 0 \\
0 & Q_{\mathbf{b}} A^2(f(s)) Q_{\mathbf{b}}
\end{pmatrix}
\end{equation}
As per the analysis in the Supplementary Material of \cite{PhysRevLett.122.060504}, the spectra of $A(f(s)) Q_{\mathbf{b}} A(f(s))$ and $Q_{\mathbf{b}} A^2(f(s)) Q_{\mathbf{b}}$ are identical.
Moreover, following that analysis, the gap of $A(f(s)) Q_{\mathbf{b}} A(f(s))$ is lower bounded by the minimum eigenvalue of $A^2(f(s))$.
In this case, since
\begin{equation}
A^2(f) =
\begin{pmatrix} (1-f)^2I+f^2 A A^\dagger & 0 \\
0 & (1-f)^2I+f^2 A^\dagger A
\end{pmatrix},
\end{equation}
the minimum eigenvalue is $(1-f)^2+(f/\kappa)^2$.
This translates to a minimum gap of $H(s)$ of $\sqrt{(1-f(s))^2+(f(s)/\kappa)^2}$ as in the Hermitian case.
By using the qubitised quantum walk for the implementation of $W_T$, we can avoid the logarithmic factor in the complexity that arises from using the Dyson series to simulate continuous Hamiltonian evolution.
In order to block encode the Hamiltonian $H(s)$, one can use block encodings of both $H_0$ and $H_1$, supplemented with an ancilla qubit that will be rotated to select between $H_0$ and $H_1$. The rotation is given by
\begin{equation}
\label{eq:C-rot}
R(s)=\frac{1}{\sqrt{\left(1-f(s)\right)^2+f(s)^2}}\begin{pmatrix} 1-f(s) & f(s)\\
f(s) & -(1-f(s))
\end{pmatrix}.
\end{equation}
To block encode $A(f(s))$, instead of using symmetric rotations, we use the initial rotation $R(s)$, then apply the controlled operations
\begin{equation}
\textsc{sel} = \ket{0}\bra{0} \otimes U_0 + \ket{1}\bra{1} \otimes U_1,
\end{equation}
where $U_0$ and $U_1$ are unitaries used for the block encodings of $\sigma_z\otimes I_N$ and $\mathbf{A}$.
Then after this operation, instead of applying the inverse of $R(s)$, we simply perform a Hadamard.
This means that, instead of block encoding $A(f(s))$, we have block encoded
\begin{equation}
\frac 1{\sqrt{2[(1-f(s))^2+f(s)^2]}} A(f(s)).
\end{equation}
This prefactor is between $1/\sqrt{2}$ and 1, and will reduce the gap.
For the qubitisation it is important that the block encoding is symmetric.
For the complete qubitisation of $H(s)$ as in \cref{eq:Hsencoding}, there are two blocks, one with $Q_\mathbf{b}$ followed by $A(f(s))$, and the other with the reverse order.
These blocks independently are asymmetric, but together they give a Hermitian Hamiltonian, and the block encoding is symmetric.
We can apply exactly the same procedure to account for the asymmetry between the $R(s)$ and Hadamard.
That is, for one block in $H(s)$, we can use $R(s)$ at the beginning and the Hadamard at the end, and the other we reverse the order.
The overall encoding is symmetric, as required for qubitisation.
We can similarly use this approach for combining $H_0$ and $H_1$ from \cref{eq:H0,eq:H0} for the case where $A$ is positive definite and Hermitian.
The qubitised operator $W_T(s)$ is then obtained by combining the block encoding of $H(s)$ with a reflection on the control qubits.
For a complete description of the procedure, see \cref{app:blockHs}.
\subsection{Choosing values for \texorpdfstring{$c_1(s)$}{c1(s)} and \texorpdfstring{$c_2(s)$}{c2(s)}}
Now, to apply \cref{cor:adia} for the QLSP two things that should be estimated are the functions $c_1(s)$ and $c_2(s)$, which in turn require upper bounds for $DW_T(s)$ and $D^{(2)}W_T(s)$.
In order to bound the difference in $W_T(s)$, we can use the fact that the only way $W_T(s)$ is dependent on $s$ is through $R(s)$.
The key feature of this operation is that it has $R(s)$ in two cross-diagonal blocks (in the matrix representation).
As a result, the spectral norm of the difference of operators is equal to the spectral norm of the difference of $R(s)$.
\begin{lemma}
\label{lem:DR}
For any $0 \leq s \leq 1-1/T$, with $W_T(s)$ encoded using the block encoding of $H(s)$ from \cref{eq:Hsencoding} together with $R(s)$ given in \cref{eq:C-rot}, it is consistent with \cref{def:difs} to choose
\begin{equation}
\label{eq:cs}
c_1(s)=2T(f(s+1/T)-f(s)),
\end{equation}
and
\begin{equation}
\label{eq:c2}
c_2(s) = \begin{cases}
2\max_{\tau\in \{s,s+1/T,s+2/T\}}\left(2|f'(\tau)|^2 + |f''(\tau)|\right), & 0 \leq s \leq 1-2/T, \\
2\max_{\tau\in \{s,s+1/T\}}\left(2|f'(\tau)|^2 + |f''(\tau)|\right), & s = 1-1/T.
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
As discussed above, to bound the difference in $W_T(s)$, we can use the difference in the rotation operation $R(s)$, which can be upper bounded as
\begin{align}
\label{eq:DR}
\left\| R(s+1/T)-R(s) \right\| &= \left\| \int_{s}^{s+1/T} \frac{dR}{ds} ds \right\| \nonumber \\
&= \left\| \int_{s}^{s+1/T} \frac{dR}{df}\frac{df}{ds} ds \right\| \nonumber \\
&\le \int_{s}^{s+1/T} \left\|\frac{dR}{df}\frac{df}{ds} \right\| ds \nonumber \\
&\le 2 \int_{s}^{s+1/T} \left|\frac{df}{ds} \right| ds \nonumber \\
&=2 \left(f(s+1/T)-f(s)\right) .
\end{align}
Here we have upper bounded the norm of $dR/df$ by 2.
To show this, the derivative of $R(s)$ with respect to $f$
\begin{equation}
\frac{dR}{df}=\frac{1}{[\left(1-f(s)\right)^2+f(s)^2]^{3/2}}\begin{pmatrix} -f(s) & 1- f(s)\\
1-f(s) & f(s)
\end{pmatrix},
\end{equation}
so the norm of $dR/df$ is $1/[(1-f(s))^2+f(s)^2]$, which varies between 1 and 2.
We have also used the fact that $df/ds>0$.
Since $\|W_T(s+1/T)-W_T(s)\|=\|R(1+1/T)-R(s)\|$ and $c_1(s)$ required in \cref{def:difs} to satisfy
\begin{equation}
\|W_T(s+1/T)-W_T(s)\| \le \frac{c_1(s)}T,
\end{equation}
we can take $c_1(s)$ as in \cref{eq:cs}.
Now for the second difference of the walk operator we use Taylor's theorem in two directions to give
\begin{equation}
W_T(s+2/T) = W_T(s+1/T) + \frac{1}{T}\frac{dW_T(s+1/T)}{ds} + \int_{s+1/T}^{s+2/T} (s+2/T-\tau)\frac{d^2W_T(\tau)}{d\tau^2}d\tau,
\end{equation}
and
\begin{equation}
W_T(s) = W_T(s+1/T) - \frac{1}{T}\frac{dW_T(s+1/T)}{ds} + \int_{s}^{s+1/T} (\tau-s)\frac{d^2W_T(\tau)}{d\tau^2}d\tau .
\end{equation}
That gives
\begin{align}
\|D^{(2)}R(s)\| &= \left\|\int_{s+1/T}^{s+2/T} (s+2/T-\tau)R''(\tau)d\tau + \int_{s}^{s+1/T} (\tau-s)R''(\tau)d\tau\right\| \nonumber \\
& \leq \frac{1}{T^2} \max_{\tau \in [s,s+2/T]}\|R''(\tau)\|.
\end{align}
Moving to the second derivative of $R(s)$ we have
\begin{equation}
\frac{d^2R(s)}{ds^2} =\frac{d^2R}{df^2}\left(\frac{df(s)}{ds}\right)^2 + \frac{dR}{df}\frac{df^2(s)}{ds^2},
\end{equation}
where
\begin{equation}
\frac{d^2R}{df^2}=\frac{1}{[\left(1-f(s)\right)^2+f(s)^2]^{5/2}}\begin{pmatrix} (4f(s)-1)f(s) -1 & (4f(s)-7)f(s)+2\\
(4f(s)-7)f(s)+2 & (1-4f(s))f(s) +1
\end{pmatrix},
\end{equation}
and its norm is
\begin{equation}
\sqrt{\frac{16(f(s)-1)f(s) +5}{\left[2(f(s)-1)f(s) +1\right]^4}},
\end{equation}
which varies between $\sqrt{5}$ and 4. Then we conclude that
\begin{equation}
\left\|D^{(2)}R\left(s\right)\right\|\leq \frac{2}{T^2}\max_{\tau\in \{s,s+1/T,s+2/T\}}\left(2|f'(\tau)|^2 + |f''(\tau)|\right).
\end{equation}
Now we have $\|D^{2}W(s)\|=\|D^{2}R(s)\|$ and \cref{def:difs} requires that $\|D^{2}W(s)\|\le c_2(s)/T^2$, so we can take $c_2(s)$ as in \cref{eq:c2}.
\end{proof}
\subsection{Linear \texorpdfstring{$\kappa$}{kappa} for \texorpdfstring{$p=3/2$}{p=3/2}}
Our next step is to show the strict linear dependence in $\kappa$ for the QLSP based on our discrete adiabatic theorem.
In the continuous case, it has been shown in~\cite{an2019quantum} that for all $1<p<2$, the corresponding AQC-based linear system solver can achieve $\mathcal{\kappa/\epsilon}$ scaling.
This suggests that taking $p$ as the midpoint of $3/2$ will give high efficiency.
Here we consider this case to estimate the constant factors in the algorithm.
\begin{theorem}[Strict linear dependence in $\kappa$]\label{theo:p15} Consider solving the QLSP $Ax=b$ for a normalised state $\ket{A^{-1}b}$, where $\|A\|=1$ and $\|A^{-1}\|=\kappa$.
By using $T \geq \kappa$ steps of a quantum walk and the schedule function of \cref{eq:sched1} with $p=3/2$,
in the case of a positive-definite and Hermitian $A$ the error in the solution may be bounded as
\begin{equation}
\label{eq:analyticWalk}
\|U_T(s) - U_T^A(s)\| \leq 5632\frac{\kappa}{T} + {\cal O}\left(\frac{\sqrt{\kappa}}{T}\right),
\end{equation}
using the encoding of $H_0$ and $H_1$ as in \cref{eq:H0,eq:H1}.
For general $A$, by using the encoding of $H(s)$ as in \cref{eq:Hsencoding} the error may be bounded by
\begin{equation}
\label{eq:analyticWalk2}
\|U_T(s) - U_T^A(s)\| \leq
15307\frac{\kappa}{T} + {\cal O}\left(\frac{\sqrt{\kappa}}{T}\right).
\end{equation}
\end{theorem}
\begin{proof}
In \cref{cor:adia} there are six terms to bound, three which are individual terms and three which are sums.
The details of the derivations of bounds on these are given in \cref{ap:upperB_cor}. Namely for the individual terms, it is shown in \cref{appsec:single} (\cref{eq:firstupperB,eq:SecondupperB,eq:thirdupperB}) that
\begin{align}
\label{eq:single_c(0)}
\frac{\hat{c}_1(0)}{T\check{\Delta}(0)^2}
&=\frac{4\sqrt{\kappa}}{T} + \mathcal{O}\left( \frac{\kappa}{T^2} \right), \\
\label{eq:single_c(1)}
\frac{\hat{c}_1(1)}{T\check{\Delta}(1)^2} &
=\frac{4\kappa}{T} + \mathcal{O}\left( \frac{\kappa}{T^2} \right), \\
\label{eq:single_c(1)_2}
\frac{\hat{c}_1(1)}{T\check{\Delta}(1)} &
=\frac{4}{T} + \mathcal{O}\left( \frac{1}{T^2} \right) ,
\end{align}
and for the sums with $c_1(s)$, it is shown in \cref{appsec:summc1} that
\begin{align}
\label{eq:sumc1}
\sum_{n=1}^{T-1}\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3}
&= \frac{16 \kappa}{T}+ \mathcal{O} \left( \frac{\kappa^{3/2}}{T^2} \right),\\
\label{eq:sumc1_2}
\sum_{n=0}^{T-1} \frac{ \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^2} &\leq \frac{16}{T} +\mathcal{O} \left( \frac {\kappa}{T^2} \right).
\end{align}
Finally for the sum with $c_2(s)$, it is shown in \cref{appsec:sumc2} that
\begin{equation}
\label{eq:sumc2}
\sum_{n=1}^{T-1} \frac{ \hat{c}_2(n/T)}{ T^2 \check{\Delta}(n/T)^2} \leq
\frac {22\kappa}{T} + \mathcal{O}\left( \frac{\kappa^{3/2}}{T^2}\right).
\end{equation}
These results are for the case where $A$ is positive definite and Hermitian.
By adding all the inequalities above, and including the constant factors in \cref{cor:adia}, we obtain the total upper bound in \cref{eq:analyticWalk}.
Note that we only include the leading term proportional to $\kappa/T$, and terms with scalings such as $\kappa^{3/2}/T^2$ are order $\sqrt{\kappa}/T$ due to the requirement that $T>\kappa$.
For the case of general $A$ (which need not be positive definite or Hermitian), the spectral gap can be lower bounded using an extra factor of $1/\sqrt 2$ as in \cref{eq:GapsA_non}.
This means that the upper bounds on the terms with $\check{\Delta}(s)$, $\check{\Delta}(s)^2$ and $\check{\Delta}(s)^3$ in the denominator may be multiplied by $\sqrt{2}$, 2 and $2\sqrt{2}$, respectively, to give new upper bounds that hold in the case of general $A$.
Adding these terms together with the weightings from \cref{cor:adia} then gives the upper bound in \cref{eq:analyticWalk2}.
There are two further subtleties in using the adiabatic algorithm for the solution.
One is that the zero eigenvalue of the Hamiltonian is degenerate, with one giving the solution, and the other just the state $\ket{b}$ but with a bit flip in an ancilla.
Because these eigenstates are orthogonal (due to the bit flip), there is no crossover between them in the adiabatic evolution.
This means that the degeneracy has no effect on the quality of the solution; see \cref{app:phasefactor}.
A further subtlety is that the qubitised quantum walk yields two eigenstates for each eigenstate of the Hamiltonian.
For the case here, the eigenvalue of the Hamiltonian we are interested in is $0$, which gives eigenvalues $\pm 1$ of the walk operator.
We may use the discrete adiabatic theorem separately on each of these eigenvalues to show that the eigenstate is preserved in the discrete adiabatic evolution.
The problem is that we need to have a positive superposition of these two eigenstates, which means that there should be no relative phase factor introduced in the adiabatic evolution.
It is shown that there is no relative phase factor in \cref{app:phasefactor}.
Therefore, neither of these subtleties has an effect on the solution, and no adjustment to the bounds is required.
\end{proof}
\subsection{General \texorpdfstring{$p$}{p}}
In this subsection, we will show that the $\mathcal{\kappa/\epsilon}$ scaling also holds for all $1<p<2$ in the discrete setting.
This result is more general, but due to a number of approximations will not be as tight an estimate as that for the specific case of $p=3/2$.
Since here we do not assume a specific value of $p$, the direct computation approach in proving \cref{theo:p15} is not applicable.
Instead, we will approximate the upper bound of the discrete error by some continuous integrals and then bound both the integrals and the approximation errors.
More precisely, we first notice that in \cref{cor:adia}, the dominant terms are the last three terms, the summations over equidistant discrete time steps.
These summations are exactly in the form of Riemann sum and approximate some integrals.
Then the dominant part of the discrete adiabatic errors can be bounded by some integrals plus the difference between the integrals and corresponding Riemann sums.
Similar to what has been shown in~\cite{an2019quantum}, the integrals exactly scale $\mathcal{O}(\kappa/T)$.
The difference between the integrals and Riemann sums is indeed of higher order according to the error bound of the first order quadrature formula.
Combining all these together, we can prove that the discrete adiabatic error for general $1<p<2$ also scales as $\mathcal{O}(\kappa/T)$, which further implies an $\mathcal{O}(\kappa/\epsilon)$ complexity of the discrete AQC-based algorithm to solve the linear system problem within $\epsilon$ error.
We summarize the main result in the following theorem.
A detailed proof is given in \cref{app:qlsp_general_p}.
\begin{theorem}[Linear dependence on $\kappa$ for general $p$]\label{thm:qlsp_general_p}
Consider using $T$ steps of the discrete adiabatic evolution with the schedule function defined in \cref{eq:sched1} for $1 < p < 2$ to solve the QLSP with general matrix $A$.
Then
\begin{enumerate}
\item for any $\kappa > 2$ and $T \geq 32d_p/3 = \mathcal{O}(\kappa^{p-1})$, there exists a positive constant $C_p$, which only depends on $p$, such that the difference between the discrete adiabatic evolution and the solution of the linear system problem can be bounded by
\begin{equation}
C_p \left(\frac{\kappa}{T}+\frac{\kappa^{p-1}}{T} + \frac{\kappa}{T^2} + \frac{1}{T}\right),
\end{equation}
\item in order to prepare an $\epsilon$-approximation of the solution of the linear system problem, it suffices to choose
\begin{equation}
T = \mathcal{O}\left(\frac{\kappa}{\epsilon}\right).
\end{equation}
\end{enumerate}
\end{theorem}
We give an explicit formula for $C_p$ in \cref{eq:Cpdef} in \cref{app:qlsp_general_p}.
We remark that \cref{thm:qlsp_general_p} only guarantees the asymptotic performance of the discrete AQC-based solvers, and the pre-constant $C_p$ is much larger than what we observe numerically.
In particular, \cref{thm:qlsp_general_p} also holds for the case when $p=3/2$, but the pre-constant in \cref{thm:qlsp_general_p} is much larger than that in \cref{theo:p15}.
This is because in \cref{thm:qlsp_general_p} we use a general proof strategy, which is applicable for all $1<p<2$ at a sacrifice of using potentially unnecessary inequalities to simplify the analysis.
These inequalities are definitely not sharp and thus result in a worse pre-constant than that obtained by direct computations in the proof of \cref{app:qlsp_general_p}.
\subsection{Numerical Results}
\begin{figure}
\caption{This figure shows the upper bound on the error in the adiabatic evolution versus the condition number $\kappa$ for a range of values of $p$ used in the scheduling function $f(s)$.
The upper bound on the error is computed using \cref{theoAdia}
\label{fig:blockerror}
\end{figure}
\begin{figure}
\caption{The upper bound on the error in the adiabatic evolution as a function of $p$ used in the scheduling function $f(s)$. In this plot we have used constant values $\kappa=40$ and $T=5\times10^4$.
\label{fig:errBlockp}
\label{fig:errBlockp}
\end{figure}
We first report the numerical results for the case where $A$ is a Hermitian and positive matrix. Rather than using the upper bounds for the first and second differences of the walk operator from \cref{lem:DR}, we exactly compute the norm of $DR$ and $D^{(2)}R$ in order to give values of $c_1(s)$ and $c_2(s)$ in \cref{theoAdia}.
We also account for the fact that the gap actual gap for the quantum walk operator is the $\arcsin$ of that in \cref{eq:Gaps}.
In~\cref{fig:blockerror} we show the numerical results for the (upper bound on the) error as a function of the condition number $\kappa$ of the matrix $A$.
We use a fixed number of steps $T=5\times 10^4$ and three different values of $p$ for the schedule function of \cref{eq:sched1}.
In each case it can be seen that the error is approximately linear in $\kappa$, which is what results in an overall complexity which is linear in $\kappa$.
The different values of $p$ result in different scaling constants with values close to 1 or 2 giving poorer scaling, which is as expected since we require $1<p<2$.
To more clearly see the dependence of the error in $p$,
in \cref{fig:errBlockp} we show the error as a function of $p$ for constant $\kappa$ (of 40).
In this case it turns out that the smallest error is for $p=1.3$, which is on the lower side of the range $(1,2)$, and smaller than the value $p=3/2$ chosen for \cref{thm:qlsp}.
From \cref{fig:errBlockp} we can also estimate the constant factors for the $\kappa/T$ scaling of the error.
In the case with $p=3/2$, for instance, as was used in \cref{theo:p15}, we have $\|U_T(s)-U_T^A(s)\|\lesssim 638 \kappa/T$. The estimate of the constant factor in \cref{theo:p15} is around 9 times bigger.
This is not unreasonable considering the many approximations made, though it indicates that the constant factor in the analysis can be improved by a more careful analysis.
\section{Filtering for solving linear equations}
\label{sec:filter}
To provide a solution to linear equations using the adiabatic method, one can use the approach of \cite{Lin2020optimalpolynomial} where the initial adiabatic algorithm is used to find the solution to some constant error (independent of $\epsilon$), then the solution can be filtered.
The approach used in \cite{Lin2020optimalpolynomial} was to apply filtering by singular value processing (similar to quantum signal processing), which is efficient and only needs one ancilla qubit, but has the drawback that it requires a highly complicated procedure for finding the correct rotation angles.
Here we provide a method using a linear combination of unitaries with similar efficiency, and only requiring two ancilla qubits (one more than singular value processing).
This has the advantage that determining the sequence of gates needed is much simpler.
The filtering by a linear combination of unitaries is similar in principle to measuring the eigenvalue of the Hamiltonian to ensure that the system is still in the ground state.
However, that does not achieve quite what we want, because it will typically produce an estimate different from the required eigenvalue.
Using a linear combination of unitaries, it may be chosen such that the ``success'' case is obtained with high probability, and one then need only consider the amplitude for incorrect states in the final state.
This is related to the principle of using symmetric states in a linear combination of unitaries.
A phase measurement would be equivalent to using preparation with the desired amplitudes at the beginning, then inverse preparation on an equal superposition at the end.
In contrast, the approach that maximises the success probability for a linear combination of unitaries is to use symmetric preparation before and after the controlled operations.
Calling the desired weights $w_j$, we would initially prepare the control register in the state
\begin{equation}\label{eq:symmsta}
\frac 1{\sqrt{\sum_j w_j}} \sum_j \sqrt{w_j} \ket{j} .
\end{equation}
Given that we are performing $j$ steps of the walk, and the input system state is an eigenvector of the walk with eigenvalue $e^{i\phi}$, the resulting state is
\begin{equation}
\frac 1{\sqrt{\sum_j w_j}} \sum_j \sqrt{w_j} e^{ij\phi} .
\end{equation}
Then projecting on the same state as in \cref{eq:symmsta} gives
\begin{equation}
\frac 1{\sum_j w_j} \sum_j w_j e^{ij\phi} .
\end{equation}
In practice, the target register will be a superposition of the eigenstates
\begin{equation}
\sum_k \psi_k \ket{k},
\end{equation}
where we are using $\ket{k}$ to indicate the eigenstate of $W_T(1)$ corresponding to eigenvalue $\phi_k$.
The state after applying the linear combination of unitaries is then
\begin{equation}
\frac 1{\sum_j w_j} \sum_{j,k} w_j \psi_k e^{ij\phi_k} \ket{k} =
\sum_{k} \tilde w(\phi_k) \psi_k \ket{k},
\end{equation}
where
\begin{equation}\label{eq:tildew}
\tilde w(\phi)= \frac 1{\sum_j w_j} \sum_{j} w_j e^{ij\phi}.
\end{equation}
Note that the state is not normalised, with the norm giving the probability of the success of this linear combination of unitaries.
We aim to have $\tilde w(\phi)$ for $\phi$ in the spectrum of interest.
Now, let us assume that the initial probability of the state on the spectrum of interest is at least $1/2$.
One can then show that the resulting normalised state obtained after the filtering has error, as quantified by the norm of the difference of states, upper bounded by
\begin{equation}
\max_{k\in\{\perp\}} \tilde w(\phi_k).
\end{equation}
where $\perp$ is the set of $k$ such that $\phi_k$ is not in the spectrum of interest (so $e^{i\phi_k}\not\in \sigma_P$).
See \cref{ap:filtering} for the proof.
The result of this reasoning is that to bound the error in the filtering, we need to bound the maximum of $\tilde w(\phi_k)$, which is minimised by the Dolph-Chebyshev window.
This is obtained by taking the discrete Fourier transform of the Chebyshev polynomials, so that $\tilde w(\phi)$ is given by Chebyshev polynomials in a similar way as for \cite{Lin2020optimalpolynomial}.
In particular, one can take
\begin{equation}
\tilde w(\phi) = \epsilon T_{\ell}\left( \beta \cos\left( \phi \right) \right)
\end{equation}
for $\phi$ taking discrete values $\pi k/\ell$ for $k$ from $-\ell$ to $\ell$, and
where $\beta=\cosh(\tfrac 1\ell \cosh^{-1}(1/\epsilon))$.
Taking the discrete Fourier transform of these values gives the window, and the Fourier transform simply yields the formula for $\tilde w(\phi)$ in terms of Chebyshev polynomials.
One obtains powers of $e^{2i\phi}$ from $-\ell/2$ to $+\ell/2$, which means we need a maximum power of $e^{i\phi}$ of $\ell$.
One can obtain the positive and negative powers simultaneously with negligible cost by simply controlling whether the reflection is performed in the qubitisation.
As a result, the cost in terms of calls to the block-encoded matrix is $\ell$, as compared to $2\ell$ for the singular value processing approach.
The peak for $\tilde w(\phi)$ will be at $0$ and $\pi$, which is what is needed because the qubitised operator produces duplicate eigenvalues at $0$ and $\pi$.
The width of the operator can be found by noting that the peak is for the argument of the Chebyshev polynomial equal to $\beta$, and the width is where the argument is 1, so $\beta\cos(\phi)=1$.
This gives us
\begin{equation}\label{eq:chebwid}
\cosh(\tfrac 1\ell \cosh^{-1}(1/\epsilon)) \cos(\phi) = 1 .
\end{equation}
Now, because the width of the peak should be equal to the gap, and the gap is $1/\kappa$, we can replace $\phi$ with $1/\kappa$, and solving for $\ell$ gives
\begin{equation}
\ell = \frac{\cosh^{-1}(1/\epsilon)}{\cosh^{-1}(1/\cos(1/\kappa))} \le \kappa \ln(2/\epsilon).
\end{equation}
Note that \cref{eq:chebwid} was for finding the width given an integer $\ell$, but solving for $\ell$ with a width of $1/\kappa$, we should round $\ell$ up to the nearest integer to provide a width no larger than $\kappa$.
In comparison, in \cite{Lin2020optimalpolynomial} the error is given as $2 e^{-\sqrt{2} \ell \Delta}$, which would imply that one can take $\ell \approx \sqrt{1/2} \kappa \ln (2/\epsilon)$.
Since the order of the polynomial is $2\ell$, which is also the number of applications of the block encoding needed, this would imply a cost of $\sqrt{2} \kappa \ln (2/\epsilon)$, which is greater than what we have here by a factor of $\sqrt{2}$.
However, it turns out that the scaling given in \cite{Lin2020optimalpolynomial} is overly conservative, and the actual scaling is $2 e^{-2 \ell \Delta}$, which then gives the same complexity as we have here.
Next we consider how to apply the linear combination of unitaries with minimum ancilla qubits.
To do this we first represent the control registers in unary.
That is, for each of the $\ell$ controlled operations, we use a single qubit which is one or zero depending on whether this operation is to be performed or not.
It may seem counterproductive to expand the size of the ancilla in this way, but it has the advantage that it has a simple state preparation procedure, where an initial qubit is rotated, then the following qubits are prepared by controlled rotations.
When doing this procedure, we can apply a just-in-time preparation procedure, where each qubit is prepared just as it is needed to be used as a control.
An example of this is shown in Fig.~\ref{fig:lcu1}.
\begin{figure}
\caption{A linear combination of steps using control registers prepared in unary using a linear sequence of controlled rotations.
\label{fig:lcu1}
\label{fig:lcu1}
\end{figure}
\begin{figure}
\caption{A linear combination of steps using control registers prepared in unary using a linear sequence of controlled rotations, but with the inverse preparation performed with the linear sequence in the reverse order.
\label{fig:lcu2}
\label{fig:lcu2}
\end{figure}
\begin{figure}
\caption{A linear combination of steps using control registers prepared in unary, but with the order of the operations changed so we only need to use two ancilla qubits at a time.
\label{fig:lcu3}
\label{fig:lcu3}
\end{figure}
Then in inverting the preparation, one could simply perform the reverse of all the controlled rotations as in Fig.~\ref{fig:lcu1}.
However, the trick is that the sequential state preparation procedure for the unary can be performed from either end.
The preparation could be achieved by performing rotations starting from the lart qubit, and working back to the first.
We do not do that for the preparation, but we do the reverse of that for the inverse preparation.
An example of this is shown in Fig.~\ref{fig:lcu2}.
When you reverse that form of preparation, you are working form the first qubit to the last, the same as for the preparation.
That means you only need to use two ancillas at once, by rearranging the operations as shown in Fig.~\ref{fig:lcu3}.
See \cref{ap:filtering} for a more explicit description of the sequence of rotations.
The major advantage of this procedure over singular value processing or quantum signal processing is that there is a very simple prescription for finding the sequence of operations.
A second advantage is that, instead of the measurement being performed at the end, measurements are performed sequentially, and a failure (the incorrect measurement result) can be flagged early.
That means that in cases where there will be a failure, it will on average be flagged halfway through, with the result that half the number of operations are needed since one can discard the state and start again.
Combining our result for the solution of QLSP with the filtering, we find that the overall complexity of the QLSP algorithm can be given as $\mathcal{O}(\kappa\log(1/\epsilon))$.
In particular, the result is as follows.
\begin{theorem}[QLSP with linear dependence on $\kappa$]\label{thm:qlsp}
Let $Ax=b$ be a system of linear equations, where $A$ is an $N$-by-$N$ matrix with $\norm{A}=1$ and $\norm{A^{-1}}=\kappa$. Given an oracle block encoding the operator $A$ and an oracle preparing $\ket{b}$, there exists a quantum algorithm which produces the normalized state $\ket{A^{-1}b}$ to within error $\epsilon$ using a number
\begin{equation}
\mathcal{O}(\kappa\log(1/\epsilon))
\end{equation}
of oracle calls.
\end{theorem}
\begin{proof}
In this theorem, we use standard assumptions that access to the oracles includes forward, reverse, and controlled uses.
We initially apply the oracle for preparing $\ket{b}$ to prepare the initial state of the form in \cref{eq:bvec}.
This preparation is also used to construct the projection operator $Q_\mathbf{b}$.
Together with the oracle for block encoding $A$, we can construct the operator for block encoding $H(s)$ as described in detail in \cref{app:blockHs}.
A reflection on the ancillas yields the walk operator.
Now use the discrete adiabatic theorem for the QLSP as given in \thm{qlsp_general_p} for fixed precision, such as $1/2$.
That step has complexity $\mathcal{O}(\kappa)$, and the only error is the overlap with other states that are not the solution.
Next, use the filtering as described above, which has complexity $\mathcal{O}(\kappa\log(1/\epsilon))$.
In the case of success, one has produced the state $\ket{A^{-1}b}$ to within norm-distance $\epsilon$.
In the case of failure of the filtering, repeat the procedure.
Since the probability of success may be made at least $1/2$ by suitably choosing the fixed precision for the adiabatic procedure, the adiabatic and filtering steps need only be applied 2 times on average before success.
This gives a factor of $2$ to the total complexity of $\mathcal{O}(\kappa)$ plus $\mathcal{O}(\kappa\log(1/\epsilon))$.
The total complexity is therefore $\mathcal{O}(\kappa\log(1/\epsilon))$ as claimed.
\end{proof}
Perhaps surprisingly, in this complexity the largest asymptotic complexity is for the filtering step, because it has a factor of $\log(1/\epsilon)$ which is absent from the adiabatic step.
In practice, we have found quite large constant factors for the adiabatic evolution, so it is likely that the adiabatic step will still be the most costly part of the algorithm for realistic values of the parameters.
In particular, for the numerical calculation of the upper bound it was found that the scaling constant was about 638, so to obtain our requirement of initial probability on the spectrum of interest at least $1/2$ (corresponding to needing to repeat the algorithm twice on average) one would need about $834\kappa$ steps of the adiabatic evolution.
In contrast, the $\ln(2/\epsilon)$ factor is only about 20 for $\epsilon$ as one part in a billion.
\section{Conclusions}
In this work we have shown the first QLSP algorithm that scales optimally in terms of the condition number. We achieved this by adapting prior algorithms for the QLSP based on adiabatic evolutions so that they did not require the additional overhead of the Dyson series algorithm for precisely evolving under time-dependent Hamiltonians on a gate model quantum computer. Instead, we show that one can directly discretize the time-evolution using quantum walks and that the error in this procedure can be obtained using a discrete adiabatic theorem. We also obtain rigorous new error bounds on the performance of those discrete adiabatic theorems.
While this improvement is ``only'' by a log factor, the fact that we can asymptotically match the lower bound is of fundamental interest. Furthermore, there is widespread anticipation that compelling practical application of the QLSP may eventually be found and that error-corrected quantum computers capable of realizing those applications may eventually be realized. Should this occur, then it will be crucial to program those devices using the best possible scaling versions of these algorithms in order to have the fastest implementations requiring the least overhead due to error-correction. Our expectation is that the QLSP approach described in this paper would be more performant than any other approach in the literature both in terms of asymptotic scaling but also in terms of the constant factors associated with realizing finite instances. Thus, we also foresee practical value in these results.
As well as scaling optimally in the condition number, our algorithm scales optimally in terms of the combination of the condition number and the precision $\epsilon$.
As was recently proven, a lower bound to the complexity is $\mathcal{O}(\kappa \log(1/\epsilon))$ \cite{RobinAram}.
Our result matches this lower bound, showing that it is optimal.
It is interesting that the complexity is multiplicative between $\kappa$ and $\log(1/\epsilon)$, in contrast to Hamiltonian simulation which is additive between the time and $\log(1/\epsilon)$.
In this approach to solving linear equations, the $\log(1/\epsilon)$ factor only comes from the filtering step, which in practice would have lower complexity than the initial adiabatic step.
Another question is the scaling with the sparsity in the case where the matrix is sparse and given by oracles for positions of nonzero entries.
In this work we have given the complexity in terms of calls to a block encoding of the matrix, rather than those more fundamental oracles.
The lower bound in terms of those oracles has a multiplicative factor of $\sqrt{d}$ in the sparsity $d$.
One could get such a scaling if there were a way of block encoding the matrix with complexity $\sqrt{d}$, but standard methods are linear.
It is shown in \cite{rootd} how to simulate a Hamiltonian with complexity $\sqrt{d}$ up to logarithmic factors using a nested interaction picture approach.
One could use that combined with the adiabatic approach to obtain this scaling with sparsity, but it would reintroduce logarithmic factors, so the complexity would no longer be strictly linear in $\kappa$.
More generally, we expect that other quantum algorithms based on continuous time-evolutions might benefit from using discrete time adiabatic algorithms. For example, there are quantum algorithms for optimization that use adiabatic evolution.
There was some analysis demonstrating that discrete adiabatic evolution could be used in \cite{Sanders2020}, but our analysis here is far tighter.
There was also recent work showing that digital adiabatic simulation based on Trotter-type formulas is robust against discretization \cite{Changhao2021}, whereas our approach does not introduce any discretization error since we directly invoke the discrete adiabatic theorem.
Our analysis here could be tightened further in terms of the constant factors.
There is over an order of magnitude difference between the numerical results and the analytically proven scaling constants.
A more careful accounting for the inequalities could tighten this difference, but we have not done that in this work because our analysis is already very lengthy.
\begin{thebibliography}{26}
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\appendix
\section{List of variables}
Here we give a list of variable names with links to their definitions.
\subsection{List of variables presented in \texorpdfstring{\cref{sec:summary}}{Section II}}
\begin{itemize}
\setlength\itemsep{0em}
\item $W_T(s)$ - The discrete walk operator.
\item $n$ - An integer index used for the discrete walk operators, so $s=n/T$.
\item $U_T(s)$ - The product of walk operators up to $s$.
item $P_T(s)$ - The projector onto the spectrum of interest.
\item $Q_T(s)$ - The projector onto the complementary spectrum.
\item $U_T^A(s)$ - The ideal adibatic evolution, in contrast to $U_T(s)$ given by the actual walk operators.
\item $W_T^A(s)$ - Ideal adiabatic walk operators that exactly preserve eigenstates.
\item $s$ - A variable used to index adiabatic evolution, starting from 0 and ending at 1.
\item $T$ - An integer corresponding to the number of discrete walk operators in discrete adiabatic evolution.
\item $T^*$ - A lower bound used for $T$ for the definition in \cref{def:gaps}.
\item $D$ - A difference operator, so for example $DW_T(s) = W_T(s+1/T)-W_T(s)$.
\item $D^{(k)}$ - The iterated difference operator.
\item $c_k(s)$ - A bound on the norm of $D^{(k)}W_T(s)$ as in \cref{def:difs}.
\item $\hat c_k(s)$ - The maximum of $c_k(s)$ over neighbouring time steps, as in \cref{eq:chat}.
\item $\sigma_P(s)$ - The spectrum of interest.
\item $\sigma_Q(s)$ - The complementary spectrum.
\item $\sigma_P^{(k)}$ - An arc including the spectrum $\sigma_P(s)$ at $k+1$ successive steps, as in \cref{eqn:assump2}.
\item $\sigma_Q^{(k)}$ - Similar to $\sigma_P^{(k)}$, but for the complementary spectrum.
\item $\Delta_k(s)$ - The gap between the spectra accounting for $k+1$ successive steps; see \cref{def:gaps}.
\item $\Delta(s)$ - The gap accounting for up to 3 successive steps as defined in \cref{eq:minGaps}.
\item $\check \Delta(s)$ - The maximum of $\Delta(s)$ accounting for neighbouring steps; see \cref{eq:fhat}.
\end{itemize}
\subsection{List of variables presented in \texorpdfstring{\cref{sec:adtheo}}{Section III}}
\begin{itemize} \setlength\itemsep{0em}
\item $R_T(s,z)$ - The resolvent of $W_T(s)$; see \cref{eq:resolv}.
\item $S_T(s,s')$ - The operator exactly mapping from the spectrum at step $s'$ to $s$; see \cref{eq:STdef}.
We also use $S_T(s)=S_T(s+1/T,s)$.
\item $V_T(s,s')$ - The unitary obtained from a polar decomposition of $S_T(s,s')$; see \cref{eq:V}.
We also use $V_T(s)=V_T(s+1/T,s)$.
\item $v_T(s,s')$ - The correction to obtain $V_T(s,s')$ from $S_T(s,s')$.
We also use $v_T(s)=v_T(s+1/T,s)$.
\item $\Omega_T(s)$ - The wave operator, accounting for the difference between the ideal and adiabatic walk; see \cref{eq:waveOp}.
\item $\Theta_T(s)$ - The ripple operator, corresponding to a step of $\Omega_T(s)$; see \cref{eq:ripple}.
\item $K_T(s)$ - The kernel function, see \cref{eq:K}.
\item $X(s)$ - Given by $T(1-V_T^{\dagger}(s-1/T))$ and used in the proof of the adiabatic theorem.
\item $\tilde X(s)$ - Obtained from a contour integral of $X(s)$ as in \cref{eq:Xtilde}.
\item $A(s)$ - A variable used in the proof of the discrete adiabatic theorem; see \cref{eq:A}.
\item $B(s)$ - Used in the proof of the discrete adiabatic theorem; see \cref{eq:B}.
\item $Z(s)$ - Used in the proof of the discrete adiabatic theorem; see \cref{eq:Z}.
\item $\Gamma_T(s)$ - A contour that encloses the spectrum of interest.
\item $\Gamma_T(s,k)$ - A contour that encloses the spectrum of interest for $k+1$ successive steps of the walk.
\item $\mathcal{F}_T(s)$ - A function of $DP_T(s)$ used for expressing $V_T(s)$; see \cref{eq:def_F}.
\item $\mathcal{B}$ - The boundary term used in \cref{lem:sum_by_parts}.
\item $\mathcal{S}$ - The sum used in \cref{lem:sum_by_parts}.
\item $n_\pm$ - We use $n_+=n+1$ and $n_-=n-1$. We also use this notation for $l$.
\item $P_0$ - The initial projector onto the spectrum of interest, $P_0=P_T(0)$.
\item $Q_0$ - Similarly for the complementary spectrum $Q_0=Q_T(0)$.
\item $\mathcal{D}_j(x)$ - The simple scalar functions $\mathcal{D}_1(x),\mathcal{D}_2(x),\mathcal{D}_3(x)$ are defined in \cref{eq:D_i}.
\item $\xi_j$ - Constants used for upper bounds on $\mathcal{D}_j(x)$ as in \cref{eq:upperB_D}.
\item $\mathcal{G}_{T,j}(s)$ - These functions for $j=1,2,3,4$ are defined in \cref{eq:G1,eq:G2,eq:G3,eq:G4}.
\end{itemize}
\subsection{ List of variables presented in \texorpdfstring{\cref{sec:linsys}}{Section IV}}
\begin{itemize} \setlength\itemsep{0em}
\item $A$ - The matrix in the QLSP $Ax=b$.
\item $b$ - The vector in the QLSP.
\item $x$ - This is usually used as the solution vector in $Ax=b$, but in \cref{ap:proofDP} as a real variable of integration.
\item $N$ - The dimension of the QLSP.
\item $\kappa$ - The condition number of $A$.
\item $\epsilon$ - The allowable error in the solution.
\item $H_0$ - The initial Hamiltonian in adiabatic evolution.
\item $H_1$ - The final Hamiltonian in adiabatic evolution.
\item $\ket{b}$ - The state with amplitudes proportional to the entries of $b$.
\item $Q_b$ - The projector eliminating $\ket{b}$, given as $I_N-\ketbra{b}{b}$.
\item $f(s)$ - Used for the scheduling function, which we take as in \cref{eq:sched1}.
\item $d_p$ - A constant used in constructing $f(s)$; see \cref{eq:gapCon}.
\item $p$ - An adjustable parameter used in the scheduling function, taking values in the range $(1,2]$.
\item $\mathbf{A}$ - A matrix constructed from $A$ so as to be Hermitian; see \cref{eq:Avec}.
\item $\mathbf{b}$ - A vector comprised of $b$ and a zero vector; see \cref{eq:bvec}.
\item $A(f)$ - The intermediate value of $A$ used in the adiabatic evolution; see \cref{eq:Af}.
\item $H(s)$ - The Hamiltonian constructed from $A(f)$; see \cref{eq:Hsencoding}.
\item $R(s)$ - A rotation used in block encoding $H(s)$; see \cref{eq:C-rot}.
\end{itemize}
\subsection{ List of variables presented in \texorpdfstring{\cref{sec:filter}}{Section V}}
\begin{itemize} \setlength\itemsep{0em}
\item $w_j$ - Weights used for the linear combination of unitaries for filtering.
\item $\phi_k$ - Used to label eigenvalues of the walk operator, so the eigenvalue is $e^{i\phi_k}$.
\item $\tilde w(\phi)$ - A Fourier transform of $w_j$ as in \cref{eq:tildew}.
\item $\perp$ - A set of $k$ such that $\phi_k$ is not in the spectrum of interest.
\item $T_\ell$ - The Chebyshev polynomial of the first kind.
\item $\ell$ - The order of the Chebyshev polynomial.
\end{itemize}
\section{Proof of \texorpdfstring{\cref{lem:P}}{Lemma 6}}\label{ap:proofDP}
In order to bound $DP_T(s)$, we first rewrite $D R_T(s,z)$ as
\begin{align}
\label{eq:contProj}
D R_T(s,z) &= \left(W_T\left(s+\frac{1}{T}\right)-z I\right)^{-1}-\left(W_T\left(s\right)-z I\right)^{-1} \nonumber\\
&= \left(W_T\left(s+\frac{1}{T}\right)-z I\right)^{-1}\left(W_T\left(s\right)-z I\right)\left(W_T\left(s\right)-z I\right)^{-1}\nonumber\\
&\quad -\left(W_T\left(s+\frac{1}{T}\right)-z I\right)^{-1}\left(W_T\left(s+\frac{1}{T}\right)-z I\right)\left(W_T\left(s\right)-z I\right)^{-1}\nonumber\\
&= \left(W_T\left(s+\frac{1}{T}\right)-z I\right)^{-1}\left(W_T\left(s\right)-W_T\left(s+\frac{1}{T}\right) \right)\left(W_T\left(s\right)-z I\right)^{-1}\nonumber\\
&= - R_T\left(s+\frac{1}{T},z\right)DW_T\left(s\right) R_T\left(s,z\right).
\end{align}
Using this expression, we can then express $DP_T(s)$ in terms of a contour integral as
\begin{align}
\label{eq:contProj2}
DP_T(s)&= \frac{1}{2 \pi i} \oint_{\Gamma_T(s,1)} \left[R_T\left(s+\frac{1}{T},z\right)-R_T(s,z)\right] dz \nonumber \\
&= -\frac{1}{2 \pi i} \oint_{\Gamma_T(s,1)} R_T\left(s+\frac{1}{T},z\right)DW_T\left(s\right) R_T\left(s,z\right)dz.
\end{align}
When we consider $P_T(s)$, the integrand drops off as $1/|z|$, so the contour must be at a finite distance as illustrated in \cref{fig:contour_1_infty}.
For $DP_T(s)$, we can use the same contour for both $P_T(s)$ and $P_T(s+1/T)$.
The principle now is that the integrand falls off as $1/|z|^2$, so the contribution from the arc will fall to zero for large radius.
We denote the contour as $\Gamma_T(s,1,a)$, which is a sector of radius $(a+1)$ for some real number $a$, and we will take the limit $a \rightarrow \infty$.
Then we have
\begin{align}
\|DP_T(s)\|
&= \frac{1}{2\pi}\left\|\oint_{\Gamma_T(s,1,a)} R_T\left(s+\frac{1}{T},z\right)DW_T\left(s\right) R_T\left(s,z\right)dz\right\| \nonumber\\
&\leq \frac{1}{2\pi}\oint_{\Gamma_T(s,1,a)} \left\|R_T\left(s+\frac{1}{T},z\right)\right\|\|DW_T\left(s\right) \|\|R_T\left(s,z\right)\||dz| \nonumber\\
&\leq \frac{c_1(s)}{T}\frac{1}{2\pi}\oint_{\Gamma_T(s,1,a)} \left\|R_T\left(s+\frac{1}{T},z\right)\right\|\| R_T\left(s,z\right)\||dz|,
\end{align}
where in the last line we used the bound from \cref{eq:main_ass}.
\begin{figure}
\caption{The contour $\Gamma_T(s,1,a)$ that passes through the two gaps, and has a closure of the contour via an arc at radius $a+1$.
The centres of the two gaps are denoted $g_{1,1}
\label{fig:contour_1_infty}
\end{figure}
Since $R_T(s,z)$ is the resolvent of the unitary operator $W_T(s)$, we know that
\begin{equation}
\label{eq:dist}
\left\|\left(W_T(s) - zI\right)^{-1}\right\|
=
\frac{1}{d\left(\sigma(W_T(s)),z\right)},
\end{equation}
where $d\left(\sigma(W_T(s)),z\right)$ is the distance between the spectrum of $W_T$ and $z$. Therefore, by separating the contour integral into three parts, two of them along the radius and one along the arc, we have that
\begin{align}
& \oint_{\Gamma_T(s,1,a)} \left\|R_T\left(s+\frac{1}{T},z\right)\right\|\left\| R_T\left(s,z\right)\right\||dz| \nonumber \\
& \leq 2 \int_0^{a+1} \frac{dx}{(x-\cos(\Delta_1(s)/2))^2 + (\sin(\Delta_1(s)/2))^2}
+ \int_{\text{arg}(g_{1,1}(s))}^{\text{arg}(g_{2,1}(s))} \frac{1}{a^2} (a+1)d\theta \nonumber \\
& \leq 2 \int_0^{a+1} \frac{dx}{(x-\cos(\Delta_1(s)/2))^2 + (\sin(\Delta_1(s)/2))^2}
+ \frac{2\pi(a+1)}{a^2}.
\end{align}
Here we have denoted the complex numbers in the centres of the gaps by $g_{1,1}(s)$ and $g_{2,1}(s)$.
By taking the limit $a \rightarrow \infty$, we have
\begin{align}\label{eq:CountInt1}
\lim_{a\to \infty} \oint_{\Gamma_T'(s,1,a)} \left\|R_T\left(s+\frac{1}{T},z\right)\right\|\left\| R_T\left(s,z\right)\right\||dz|
&\le 2\int_0^{\infty}\frac{dx}{(x-\cos{\left(\Delta_1(s)/2\right)})^2+\sin{\left(\Delta_1(s)/2\right)^2}} \nonumber \\
&= \frac{2\pi-\Delta_1(s)}{\sin{\left(\Delta_1(s)/2\right)}} \nonumber \\
& \le \frac{4\pi}{\Delta_1(s)},
\end{align}
where in the last line we have used $(\pi-x)/\sin x \leq \pi/x$ for $0 < x \leq \pi/2$.
Note that taking the limit of $a\to\infty$ the contribution from the arc completely vanishes, and we have integrals to infinity along the two straight lines for the contour.
That gives a bound on $\|DP_T(s)\|$ as
\begin{equation}
\label{eq:CountInt}
\|DP_T(s)\|\leq \frac{2c_1(s)}{T\Delta_1(s)}.
\end{equation}
A number of other integrals that can be obtained in a similar way.
In exactly the same way, we have
\begin{align}\label{eq:CountInt2}
\oint_{\Gamma_T(s,1)} \left\|R_T\left(s+\frac{1}{T},z\right)\right\|^2\left\| R_T\left(s,z\right)\right\||dz|
&\le 2\int_0^{\infty}\frac{dx}{[(x-\cos{\left(\Delta_1(s)/2\right)})^2+\sin{\left(\Delta_1(s)/2\right)^2]^{3/2}}} \nonumber \\
&= \frac 2{1-\cos(\Delta_1(s)/2)} .
\end{align}
The same bound holds for similar products of three terms.
Here we have written the integral as for the contour $\Gamma_T(s,1)$.
This contour can be regarded as the limit as $a\to\infty$ of the contour $\Gamma_T(s,1,a)$, but from now on we will omit the explicit procedure taking the limit.
Now we move on to the $D^{(2)}P_T$. Now since we are dealing with the second order difference, the contour should be chosen to be $\Gamma_T(s,2)$ which passes through the eigenvalue gap for three consecutive steps.
The reasoning for the contour integrals above is unchanged, except the gap $\Delta_1(s)$ is changed to $\Delta_2(s)$ for three consecutive steps.
We therefore have
\begin{align}
D^{(2)}P_T(s) & = D P_T\left(s+\frac{1}{T}\right) - D P_T\left(s\right) \nonumber\\
& = -\frac{1}{2\pi i} \oint_{\Gamma_T(s,2)} \left[R_T\left(s+\frac{2}{T},z\right)DW_T\left(s+\frac{1}{T}\right) R_T\left(s+\frac{1}{T},z\right) - R_T\left(s+\frac{1}{T},z\right)DW_T\left(s\right) R_T\left(s,z\right)\right] dz \nonumber\\
& = -\frac{1}{2\pi i} \oint_{\Gamma_T(s,2)} \left[R_T\left(s+\frac{2}{T},z\right) -R_T\left(s+\frac{1}{T},z\right) \right]DW_T\left(s+\frac{1}{T}\right) R_T\left(s+\frac{1}{T},z\right) dz \nonumber\\
& \quad -\frac{1}{2\pi i} \oint_{\Gamma_T(s,2)} R_T\left(s+\frac{1}{T},z\right) D^{(2)}W_T\left(s\right) R_T\left(s+\frac{1}{T},z\right) dz \nonumber\\
& \quad -\frac{1}{2\pi i} \oint_{\Gamma_T(s,2)} R_T\left(s+\frac{1}{T},z\right)DW_T\left(s\right) \left[R_T\left(s+\frac{1}{T},z\right) -R_T\left(s,z\right) \right] dz \nonumber\\
& = \frac{1}{2\pi i} \oint_{\Gamma_T(s,2)} R_T\left(s+\frac{2}{T},z\right)DW_T\left(s\right) R_T\left(s+\frac{1}{T},z\right)DW_T\left(s+\frac{1}{T}\right) R_T\left(s+\frac{1}{T},z\right) dz \nonumber\\
& \quad -\frac{1}{2\pi i} \oint_{\Gamma_T(s,2)} R_T\left(s+\frac{1}{T},z\right) D^{(2)}W_T\left(s\right) R_T\left(s+\frac{1}{T},z\right) dz \nonumber\\
& \quad -\frac{1}{2\pi i} \oint_{\Gamma_T(s,2)} R_T\left(s+\frac{1}{T},z\right)DW_T\left(s\right) R_T\left(s+\frac{1}{T},z\right)DW_T\left(s\right) R_T\left(s,z\right) dz.
\end{align}
We can bound the first term as
\begin{align}
& \left\| \frac{1}{2\pi i} \oint_{\Gamma_T(s,2)} R_T\left(s+\frac{2}{T},z\right)DW_T\left(s\right) R_T\left(s+\frac{1}{T},z\right)DW_T\left(s+\frac{1}{T}\right) R_T\left(s+\frac{1}{T},z\right) dz \right\| \nonumber \\
& \le \frac{1}{2\pi} \oint_{\Gamma_T(s,2)} \left\|R_T\left(s+\frac{2}{T},z\right)\right\|\left\|DW_T\left(s\right)\right\| \left\|R_T\left(s+\frac{1}{T},z\right)\right\|\left\|DW_T\left(s+\frac{1}{T}\right)\right\| \left\|R_T\left(s+\frac{1}{T},z\right)\right\| |dz| \nonumber \\
& \le \frac{c_1(s) c_1(s+1/T)}{T^2}\frac{1}{2\pi} \oint_{\Gamma_T(s,2)} \left\|R_T\left(s+\frac{2}{T},z\right)\right\| \left\|R_T\left(s+\frac{1}{T},z\right)\right\| \left\|R_T\left(s+\frac{1}{T},z\right)\right\| |dz| \nonumber \\
& \le \frac{c_1(s) c_1(s+1/T)}{T^2}\frac{1}{\pi} \int_0^{\infty} \frac{dx}{[(x-\cos(\Delta_2(s)/2))^2+\sin(\Delta_2(s))/2)^2]^{3/2}} \nonumber \\
& = \frac{c_1(s) c_1(s+1/T)}{\pi T^2} \frac{1}{1-\cos(\Delta_2(s)/2)}.
\end{align}
For the second term we have the upper bound
\begin{align}
&\left\| \frac{1}{2\pi i} \oint_{\Gamma_T(s,2)} R_T\left(s+\frac{1}{T},z\right) D^{(2)}W_T\left(s\right) R_T\left(s+\frac{1}{T},z\right) dz \right\| \nonumber \\
&\le \frac{1}{2\pi} \oint_{\Gamma_T(s,2)} \left\|R_T\left(s+\frac{1}{T},z\right)\right\| \left\| D^{(2)}W_T\left(s\right)\right\| \left\| R_T\left(s+\frac{1}{T},z\right)\right\| |dz| \nonumber \\
& \le \frac{c_2(s)}{T^2} \frac{1}{2\pi} \oint_{\Gamma_T(s,2)} \left\|R_T\left(s+\frac{1}{T},z\right)\right\| \left\| R_T\left(s+\frac{1}{T},z\right)\right\| |dz| \nonumber \\
& \le \frac{c_2(s)}{\pi T^2}\int_0^{\infty} \frac{dx}{(x-\cos(\Delta_2(s)/2))^2+\sin(\Delta_2(s))/2)^2} \nonumber \\
& = \frac{c_2(s)}{\pi T^2} \frac{\pi-\Delta_2(s)/2}{\sin(\Delta_2(s)/2)} \nonumber \\
& \le \frac{2c_2(s)}{T^2} \frac{1}{\Delta_2(s)} .
\end{align}
For the third term we have identical reasoning as for the first term, except the $DW_T(s+1/T)$ is replaced with $DW_T(s)$.
That gives an upper bound
\begin{equation}
\frac{c_1^2(s)}{\pi T^2} \frac{1}{1-\cos(\Delta_2(s)/2)}
\end{equation}
The three bounds together give us
\begin{equation}
\|D^{(2)}P_T(s)\| \leq \frac{ c_1(s)^2+c_1(s)c_1(s+1/T)}{\pi T^2 (1-\cos(\Delta_2(s)/2))} + \frac{2c_2(s)}{T^2\Delta_2(s)}.
\end{equation}
\section{Proof of \texorpdfstring{\cref{lem:sum_by_parts}}{Lemma 14}}\label{ap:sum_parts}
Our initial point is noticing the following identity
\begin{equation}
\label{eq:p1}
Q_T(s)X(s)P_T(s)=-Q_T(s)[W_T(s),\Tilde{X}(s)]P_T(s),
\end{equation}
which follows from
\begin{align}
[W_T(s),\Tilde{X}(s)]&= -\frac{1}{2\pi i}\oint_{\Gamma_T(s)}[W_T(s),R_T(s,z)X(s)R_T(s,z)]dz \nonumber\\
&= -\frac{1}{2\pi i}\oint_{\Gamma_T(s)}[W_T(s)-zI,R_T(s,z)X(s)R_T(s,z)]dz \nonumber\\
&= -\frac{1}{2\pi i}\oint_{\Gamma_T(s)}(X(s)R_T(s,z)-R_T(s,z)X(s))dz \nonumber\\
&= [P_T(s),X(s)].\nonumber
\end{align}
Now, using the definition \cref{eq:Wa} for $W_T^A$, one gets $W_T(s)=V_T^{\dagger}(s)W_T^A(s)$. Substituting into \cref{eq:p1}, one gets
\begin{equation}
\label{eq:p2}
Q_T(s)X(s)P_T(s)=-Q_T(s)[W^A_T(s),\Tilde{X}(s)]P_T(s)-Q_T(s)[A(s),\Tilde{X}(s)]P_T(s),
\end{equation}
where $A(s)$ is given in \cref{eq:A}. Now, we can use $P_T(s)=U_T^{A}(s)P_0U_T^{A\dagger}(s)$ and $Q_T(s)=U_T^{A}(s)Q_0U_T^{A\dagger}(s)$ in \cref{eq:p2} to obtain
\begin{equation}
\label{eq:p3}
Q_0U_T^{A\dagger}(s)X(s)U_T^{A}(s)P_0 = -Q_0 U_T^{A\dagger}(s) [W^A_T(s),\Tilde{X}(s)]U_T^{A}(s)P_0 - Q_0U_T^{A\dagger}(s)[A(s),\Tilde{X}(s)]U_T^{A}(s)P_0,
\end{equation}
then,
\begin{align}
&U_T^{A\dagger}\left(\frac{n}{T}\right)\left[W_T^{A}\left(\frac{n}{T}\right),\Tilde{X}\left(\frac{n}{T}\right)\right]U_T^A\left(\frac{n}{T}\right)\nonumber
\\&= U_T^{A\dagger}\left(\frac{n}{T}\right)W_T^{A}\left(\frac{n}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right) - U_T^{A\dagger}\left(\frac{n}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)W_T^{A}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right)\nonumber\\
&= U_T^{A\dagger}\left(\frac{n}{T}\right)W_T^{A}\left(\frac{n}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right)-U_T^{A\dagger}\left(\frac{n}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n_+}{T}\right)\nonumber \\
&= U_T^{A\dagger}\left(\frac{n}{T}\right)W_T^{A}\left(\frac{n_-}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right)-U_T^{A\dagger}\left(\frac{n}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n_+}{T}\right)\nonumber\\
&\quad+ U_T^{A\dagger}\left(\frac{n}{T}\right)\left(W_T^{A}\left(\frac{n}{T}\right)-W_T^{A}\left(\frac{n_-}{T}\right)\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right)\nonumber\\
&= U_T^{A\dagger}\left(\frac{n_-}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right)-U_T^{A\dagger}\left(\frac{n}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n_+}{T}\right)\nonumber\\
&\quad+ U_T^{A\dagger}\left(\frac{n}{T}\right)DW_T^{A}\left(\frac{n_-}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right) \nonumber \\
&= U_T^{A\dagger}\left(\frac{n_-}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right)-U_T^{A\dagger}\left(\frac{n}{T}\right)\Tilde{X}\left(\frac{n_+}{T}\right)U_T^A\left(\frac{n_+}{T}\right) \nonumber\\
& \quad + U_T^{A\dagger}\left(\frac{n}{T}\right)DW_T^{A}\left(\frac{n_-}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right) + U_T^{A\dagger}\left(\frac{n}{T}\right)D\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n_+}{T}\right),\nonumber\\
&= U_T^{A\dagger}\left(\frac{n_-}{T}\right)\Tilde{X}\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right) - U_T^{A\dagger}\left(\frac{n}{T}\right)\Tilde{X}\left(\frac{n_+}{T}\right)U_T^A\left(\frac{n_+}{T}\right) + U_T^{A\dagger}\left(\frac{n}{T}\right)B\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right),
\end{align}
where $B(s)$ is given in \cref{eq:B}.
To complete this proof we have to multiply by $Y(s)$ on the right-hand side of Eq.~\eqref{eq:p3} and then do a sum from $1/T$ to $l/T$. First let us look to the boundary term, which is derived from the first part on the right-hand side of Eq. (\ref{eq:p3}), i.e.,
\begin{align}
&-\sum_{n=1}^l Q_0U_T^{A\dagger}\left(\frac{n}{T}\right)\left[W^A_T\left(\frac{n}{T}\right),\Tilde{X}\left(\frac{n}{T}\right)\right]U_T^{A}\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right)\nonumber\\
&= \sum_{n=1}^l Q_0U_T^{A\dagger}\left(\frac{n}{T}\right) \Tilde{X}\left(\frac{n_+}{T}\right) U_T^A\left(\frac{n_+}{T}\right) P_0 Y\left(\frac{n}{T}\right) -\sum_{n=1}^l Q_0 U_T^{A\dagger}\left(\frac{n_-}{T}\right) \Tilde{X}\left(\frac{n}{T}\right) U_T^A\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right)\nonumber\\
&\quad - \sum_{n=1}^l Q_0 U_T^{A\dagger}\left(\frac{n}{T}\right)B\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right)\nonumber \\
&= \sum_{n=1}^l Q_0U_T^{A\dagger}\left(\frac{n}{T}\right) \Tilde{X}\left(\frac{n_+}{T}\right) U_T^A\left(\frac{n_+}{T}\right) P_0 Y\left(\frac{n}{T}\right) -\sum_{n=0}^{l-1} Q_0 U_T^{A\dagger}\left(\frac{n}{T}\right) \Tilde{X}\left(\frac{n_+}{T}\right) U_T^A\left(\frac{n_+}{T}\right) P_0 Y\left(\frac{n_+}{T}\right)\nonumber\\
&\quad - \sum_{n=1}^l Q_0 U_T^{A\dagger}\left(\frac{n}{T}\right)B\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right) \nonumber \\
& = \mathcal{B}- \sum_{n=1}^l Q_0U_T^{A\dagger}\left(\frac{n}{T}\right) \Tilde{X}\left(\frac{n_+}{T}\right) U_T^A\left(\frac{n_+}{T}\right) P_0 DY\left(\frac{n}{T}\right) - \sum_{n=1}^l Q_0 U_T^{A\dagger}\left(\frac{n}{T}\right)B\left(\frac{n}{T}\right)U_T^A\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right).
\end{align}
Here we have combined two sums using $Y(s)=-DY(s)+Y(s+1/T)$, and $\mathcal{B}$ is a correction accounting for the extra term at $n=0$ and the missing term at $n=l$. It is given by
\begin{equation}
\mathcal{B}=Q_0U^{A\dagger}_T\left(\frac{l}{T}\right)\Tilde{X}\left(\frac{l_+}{T}\right)U^{A}_T\left(\frac{l_+}{T}\right)P_0Y\left(\frac{l_+}{T}\right)
-Q_0U_T^{A\dagger}(0)\Tilde{X}\left(\frac{1}{T}\right)U^{A}_T\left(\frac{1}{T}\right)P_0Y\left(\frac{1}{T}\right).
\end{equation}
Then plug the result above into Eq.~\eqref{eq:p3}, to give
\begin{align}
&\sum_{n=1}^l Q_0 U_T^{A\dagger}\left(\frac{n}{T}\right) X\left(\frac{n}{T}\right) U_T^{A}\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right)\nonumber\\
& = \mathcal{B}- \sum_{n=1}^l\left\{ Q_0U_T^{A\dagger}\left(\frac{n}{T}\right) \Tilde{X}\left(\frac{n_+}{T}\right) U_T^A\left(\frac{n_+}{T}\right) P_0 DY\left(\frac{n}{T}\right)\right. \nonumber\\
&\quad+ \left. Q_0 U_T^{A\dagger}\left(\frac{n}{T}\right)\left(\left[A\left(\frac{n}{T}\right),\Tilde{X}\left(\frac{n}{T}\right)\right]+B\left(\frac{n}{T}\right)\right) U_T^A\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right)\right\},\nonumber\\
&=\mathcal{B}-\frac{1}{T}\mathcal{S},
\end{align}
where
\begin{align}
\mathcal{S}&=\sum_{n=1}^l\left\{ Q_0U_T^{A\dagger}\left(\frac{n}{T}\right) \Tilde{X}\left(\frac{n_+}{T}\right) U_T^A\left(\frac{n_+}{T}\right) P_0 T DY\left(\frac{n}{T}\right)\right.\nonumber\\
&\quad+ \left. TQ_0 U_T^{A\dagger}\left(\frac{n}{T}\right)\left(\left[A\left(\frac{n}{T}\right),\Tilde{X}(\frac{n}{T})\right]+B\left(\frac{n}{T}\right)\right) U_T^A\left(\frac{n}{T}\right) P_0 Y\left(\frac{n}{T}\right)\right\}.
\end{align}
\section{Details for the proof of \texorpdfstring{\cref{theoAdia}}{Theorem 15} and its corollary}\label{ap:theoAdia}
\subsection{Diagonal term}\label{ap:sec_diag}
Here we bound the ``diagonal'' term in \cref{eq:diag1}.
For this term (without loss of generality we only consider the term projected on $P_0$), we have
\begin{align}
&\quad \left\|\sum_{n=1}^{sT}P_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)P_0\Omega_T\left(\frac{n_-}{T}\right)\right\| \nonumber\\
& = \left\|\sum_{n=1}^{sT}{U_T^A}^{\dagger}\left(\frac{n}{T}\right)P_T\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right)P_T\left(\frac{n}{T}\right){U_T^A}\left(\frac{n}{T}\right)\Omega_T\left(\frac{n_-}{T}\right)\right\| \nonumber\\
& \leq \sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right)P_T\left(\frac{n}{T}\right)\right\|\left\|\Omega_T\left(\frac{n_-}{T}\right)\right\| \nonumber\\
& = \sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right)P_T\left(\frac{n}{T}\right)\right\|.
\end{align}
In the second line we have used \cref{eq:projU}.
Using \cref{lem:V_v2}, we have
\begin{align}
\label{eq:diag_p1}
& \quad \sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right)P_T\left(\frac{n}{T}\right)\right\| \nonumber\\
&= \sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(I - \mathcal{F}_T\left(\frac{n_-}{T}\right) + \left(2P_T\left(\frac{n_-}{T}\right)-I\right)DP_T\left(\frac{n_-}{T}\right)\mathcal{F}_T\left(\frac{n_-}{T}\right)\right)P_T\left(\frac{n}{T}\right)\right\|\nonumber\\
& \leq \sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(I - \mathcal{F}_T\left(\frac{n_-}{T}\right)\right)P_T\left(\frac{n}{T}\right)\right\| + \sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(2P_T\left(\frac{n_-}{T}\right)-I\right)DP_T\left(\frac{n_-}{T}\right)\mathcal{F}\left(\frac{n_-}{T}\right)P_T\left(\frac{n}{T}\right)\right\| \nonumber\\
& \leq \sum_{n=0}^{sT-1}\left\|I-\mathcal{F}_T\left(\frac{n}{T}\right)\right\| + \sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(2P_T\left(\frac{n_-}{T}\right)-I\right)DP_T\left(\frac{n_-}{T}\right)\left(\mathcal{F}_T\left(\frac{n_-}{T}\right) - I\right)P_T\left(\frac{n}{T}\right)\right\| \nonumber\\
& \quad + \sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(2P_T\left(\frac{n_-}{T}\right)-I\right)DP_T\left(\frac{n_-}{T}\right)P_T\left(\frac{n}{T}\right)\right\|
\end{align}
From the second term in the last inequality we have
\begin{align}
&\sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(2P_T\left(\frac{n_-}{T}\right)-I\right)DP_T\left(\frac{n_-}{T}\right)\left(\mathcal{F}_T\left(\frac{n_-}{T}\right) - I\right)P_T\left(\frac{n}{T}\right)\right\|\nonumber\\
&\leq \sum_{n=1}^{sT}\left\|DP_T\left(\frac{n_-}{T}\right)\left(\mathcal{F}_T\left(\frac{n_-}{T}\right) - I\right)\right\|\nonumber\\
&=\sum_{n=0}^{sT-1}\left\|I-\mathcal{F}_T\left(\frac{n}{T}\right)\right\| \left\| DP_T\left(\frac{n}{T}\right) \right\|.
\end{align}
Now if we replace $P_T(n_{-}/T)=P_T(n/T)-DP_T(n_{-}/T)$ in the last term of \cref{eq:diag_p1}, i.e.,
\begin{align}
\left\|P_T\left(\frac{n}{T}\right)\left(2P_T\left(\frac{n_-}{T}\right) - I\right) DP_T\left(\frac{n_-}{T}\right) P_T\left(\frac{n}{T}\right)\right\| &= \left\|P_T\left(\frac{n}{T}\right ) \left[2\left(P_T\left(\frac{n}{T}\right)-DP_T\left(\frac{n_{-}}{T}\right)\right)-I\right] DP_T\left(\frac{n_-}{T}\right)P_T\left(\frac{n}{T}\right)\right\|\nonumber\\
&\leq \left\|P_T\left(\frac{n}{T}\right ) \left(2P_T\left(\frac{n}{T}\right)-I\right) DP_T\left(\frac{n_-}{T}\right)P_T\left(\frac{n}{T}\right)\right\|\nonumber\\
&\quad+2\left\|P_T\left(\frac{n}{T}\right ) DP_T\left(\frac{n_-}{T}\right)^2 P_T\left(\frac{n}{T}\right)\right\|\nonumber\\
&= \left\|P_T\left(\frac{n}{T}\right ) DP_T\left(\frac{n_-}{T}\right)P_T\left(\frac{n}{T}\right)\right\|\nonumber\\
&\quad+ 2\left\|P_T\left(\frac{n}{T}\right ) DP_T\left(\frac{n_-}{T}\right)^2 P_T\left(\frac{n}{T}\right)\right\|\nonumber\\
&= 3\left\|P_T\left(\frac{n}{T}\right ) DP_T\left(\frac{n_-}{T}\right)^2 P_T\left(\frac{n}{T}\right)\right\|,
\end{align}
In the last calculation above, we used the following equality $p(p-q)p=p(p-q)^2p$ when we have $p,q$ as any two projections. Thus, \begin{align}
\label{eq:diag_p2}
& \quad \sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right)P_T\left(\frac{n}{T}\right)\right\| \nonumber\\
& \leq \sum_{n=0}^{sT-1} \left\|I-\mathcal{F}_T\left(\frac{n}{T}\right)\right\|\left(1+\left\|DP_T\left(\frac{n}{T}\right)\right\|\right) + 3\sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)DP_T\left(\frac{n_-}{T}\right)^2 P_T\left(\frac{n}{T}\right)\right\| \nonumber\\
& = \sum_{n=0}^{sT-1} \left\|I-\mathcal{F}_T\left(\frac{n}{T}\right)\right\|\left(1+\left\|DP_T\left(\frac{n}{T}\right)\right\|\right) + 3\sum_{n=1}^{sT}\left\|P_T\left(\frac{n}{T}\right)(DP_T\left(\frac{n_-}{T}\right)^2 P_T\left(\frac{n}{T}\right)\right\| \nonumber\\
& \leq \sum_{n=0}^{sT-1} \left\|I-\mathcal{F}_T\left(\frac{n}{T}\right)\right\|\left(1+\left\|DP_T\left(\frac{n}{T}\right)\right\|\right) + 3\sum_{n=0}^{sT-1}\left\|DP_T\left(\frac{n}{T}\right)\right\|^2.
\end{align}
To bound $\|\mathcal{F}_T(s) - I\|$, we can use \cref{lem:P} and the definition of $\mathcal{F}_T(s)$ as follows
\begin{align}
\label{eq:bounF-i}
\|\mathcal{F}_T(s) - I\| &\leq \sum_{k=1}^{\infty} \frac{\Pi_{i=1}^{k}(2i-1)}{2^k k!}\left\| DP_T(s) \right\|^{2k}\nonumber\\
&\leq \sum_{k=1}^{\infty} \frac{\Pi_{i=1}^{k}(2i-1)}{2^k k!}\left(\frac{2c_1(s)}{T\Delta_1(s)}\right)^{2k}\nonumber\\
&= \left(1-\frac{4c_1(s)^2}{T^2\Delta_1(s)^2}\right)^{-1/2}-1,\nonumber\\
&=\mathcal{D}_1\left(\frac{2c_1(s)}{T\Delta_1(s)}\right)-1.
\end{align}
\subsection{Off-diagonal term}\label{ap:offD}
For the ``off-diagonal'' term in \cref{eq:offdiag1} used for \cref{theoAdia}, we have
\begin{align}
\label{eq:offDiag_B}
& \quad \left\|\sum_{n=1}^{sT}Q_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)\left(I - V_T^{\dagger}\left(\frac{n_-}{T}\right)\right){U_T^A}\left(\frac{n}{T}\right)P_0\Omega_T\left(\frac{n_-}{T}\right)\right\| \nonumber\\
& \leq \frac{1}{T}\left\|\sum_{n=1}^{sT-1}Q_0{U_T^A}^{\dagger}\left(\frac{n}{T}\right)X\left(\frac{n}{T}\right){U_T^A}\left(\frac{n}{T}\right)P_0Y\left(\frac{n}{T}\right)\right\| + \frac{1}{T}\left\|Q_0{U_T^A}^{\dagger}\left(s\right)X\left(s\right){U_T^A}\left(s\right)P_0Y\left(s\right)\right\| \nonumber\\
& \leq \frac 1T \| \mathcal{B} \| + \frac 1{T^2} \|\mathcal{S} \| + \frac{1}{T} \left\|X\left(s\right)\right\|\left\|Y\left(s\right)\right\| \nonumber \\
& \leq \frac{1}{T}\left\|\tilde{X}\left(\frac{1}{T}\right)\right\|\left\|Y\left(\frac{1}{T}\right)\right\| + \frac{1}{T} \left\|\tilde{X}\left(s\right)\right\|\left\|Y\left(s\right)\right\|
+ \frac{1}{T^2}\sum_{n=1}^{sT-1} \left\|Z\left(\frac{n}{T}\right)\right\| \left\|Y\left(\frac{n}{T}\right)\right\|+ \frac{1}{T}\sum_{n=1}^{sT-1} \left\|\tilde{X}\left(\frac{n_+}{T}\right)\right\| \left\|DY\left(\frac{n}{T}\right)\right\| \nonumber\\
&\quad+ \frac{1}{T} \left\|X\left(s\right)\right\|\left\|Y\left(s\right)\right\| \nonumber\\
& = \frac{1}{T}\left\|\tilde{X}\left(\frac{1}{T}\right)\right\| + \frac{1}{T}\left\|\tilde{X}\left(s\right)\right\|
+ \frac{1}{T^2}\sum_{n=1}^{sT-1} \left\|Z\left(\frac{n}{T}\right)\right\| + \frac{1}{T}\sum_{n=1}^{sT-1} \left\|\tilde{X}\left(\frac{n_+}{T}\right)\right\| \left\|DY\left(\frac{n}{T}\right)\right\|+ \frac{1}{T}\left\|X\left(s\right)\right\|.
\end{align}
In the third line we have used the summation by parts result in \cref{lem:sum_by_parts} with $l=sT-1$.
In the fourth line we have used the product rule for norms of products and the fact that the spectral norms of projectors and unitary operators are 1.
In the last line we have used the fact that the choice of $Y_T$ is unitary.
Next we use the previously derived lemmas to provide bounds for the individual operators, $X(s)$, $\tilde{X}(s)$, $Z(s)$ and $DY(s)$.
Starting with $X(s)$ we have from \cref{lem:V_bound2} combined with \cref{lem:P} that
\begin{align}
\left\|X\left(\frac{n}{T}\right)\right\|&=T\left\|V\left(\frac{n_-}{T}\right)-I\right\|\nonumber\\
&\leq T\left\|\mathcal{F}_T\left(\frac{n_-}{T}\right) - I\right\| + T\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\left\|\mathcal{F}_T\left(\frac{n_-}{T}\right)\right\|.
\end{align}
Now we use the upper bound from \cref{eq:bounF-i} to provide a bound on $\mathcal{F}_T(s)$ as
\begin{align}
\|\mathcal{F}_T(s)\| &\leq 1+\sum_{k=1}^{\infty} \frac{\Pi_{i=1}^{k}(2i-1)}{2^k k!}\left(\frac{2c_1(s)}{T\Delta_1(s)}\right)^{2k}\nonumber\\
&= \left(1-\frac{4c_1(n_-/T)^2}{T^2\Delta_1(n_-/T)^2}\right)^{-1/2},
\end{align}
and similarly
\begin{align}
\|\mathcal{F}_T(s) - I\| \leq \left(1-\frac{4c_1(n_-/T)^2}{T^2\Delta_1(n_-/T)^2}\right)^{-1/2} - 1.
\end{align}
That gives the following upper bound for $X(s)$
\begin{align}\label{eq:Xbnd}
\left\|X\left(\frac{n}{T}\right)\right\|&\leq T\left[\left(1+\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right)\left(1-\frac{4c_1(n_-/T)^2}{T^2\Delta_1(n_-/T)^2}\right)^{-1/2}-1\right]\nonumber\\
&=T\left[\left(1+\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right)^{1/2}\left(1-\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right)^{-1/2}-1\right]\nonumber\\
&=T\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right) .
\end{align}
Now for $\tilde{X}(s)$ we can use the bound from \cref{lem:Xt}, which gives
\begin{align}\label{eq:Xtildebnd}
\left\|\tilde{X}\left(\frac{n}{T}\right)\right\|&\leq
\frac{2T}{\Delta_0(n/T)}\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right).
\end{align}
For the bound on $Z(s)$ we can use \cref{eq:Zbound} from \cref{lem:ABZ2_v2}, but first we need bounds for $DX(s)$ and $DV_T(s-1/T)$, which can be obtained using \cref{lem:DV_v2} in combination with \cref{lem:P} as follows
\begin{align}\label{eq:XVrel}
\left\|DX\left(\frac{n}{T}\right)\right\| &=T\left\|DV_T\left(\frac{n_-}{T}\right)\right\|\\
&\leq T\left(1+\|DP_T(n/T)\|\right)\left\|D^{(2)}P_T(n_-/T)\right\| \mathcal{D}_3\left( \max(\|DP_T(n/T)\|,\|DP_T(n_-/T)\|) \right)\nonumber\\
&\quad+T\|\mathcal{F}_T(n_-/T)\|\left(\left\|D^{(2)}P_T(n_-/T)\right\| + 2 \|(DP_T(n_-/T))\|^2\right)\nonumber\\
&\leq \left(1+\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right)\frac{\mathcal{G}_{T,1}(n_-/T)}{T}\mathcal{D}_3\left( \max\left(\frac{2c_1(n/T)}{T\Delta_1(n/T)},\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right) \right)\nonumber\\
&\quad +\mathcal{D}_1\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right)\left(\frac{\mathcal{G}_{T,1}(n_-/T)}{T} + \frac{8c_1(n_-/T)^2}{T\Delta_1(n_-/T)^2}\right).
\end{align}
In the first line we have used $X(s) = T(I- V_T^\dagger(s-1/T))$, in the second line we have used \cref{lem:DV_v2}, and at the end we have used \cref{lem:P} in combination with the fact that the functions $\mathcal{D}_1$ and $\mathcal{D}_3$ are monotonically increasing.
Now the functions $\mathcal{G}_{T,2}$ and $\mathcal{G}_{T,3}$ from
\cref{eq:G2,eq:G3} can be used in the last expression above, to give
\begin{align}
\left\|DX\left(\frac{n}{T}\right)\right\| &\leq \left(1+\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right)\frac{\mathcal{G}_{T,2}(n_-/T)}{T})\nonumber\\
&+\mathcal{D}_1\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right)\left(\frac{\mathcal{G}_{T,1}(n_-/T)}{T} + \frac{8c_1(n_-/T)^2}{T\Delta_1(n_-/T)^2}\right)\nonumber\\
&=\frac{\mathcal{G}_{T,3}(n_-/T)}{T}.
\end{align}
Then, from \cref{eq:XVrel} we have
\begin{equation}
\left\|DV_T\left(\frac{n_-}{T}\right)\right\| \leq \frac{\mathcal{G}_{T,3}(n_-/T)}{T^2}.
\end{equation}
Now that we have these bounds we proceed to bound $Z(s)$ using \cref{eq:Zbound}.
Starting with the replacement of the bound of $DV_T(s-1/T)$ we can make use of the function ${\mathcal{G}_{T,4}(s)}$ as defined in \cref{eq:G4}
\begin{align}
\left\|Z\left(\frac{n}{T}\right)\right\| &\leq \frac{4T}{\Delta_0\left(n/T\right)} \left(\left\|\mathcal{F}_T\left(\frac{n}{T}\right) - I\right\| + \left\|DP_T\left(\frac{n}{T}\right)\right\|\left\|\mathcal{F}_T\left(\frac{n}{T}\right) \right\|\right)\left\|X\left(\frac{n}{T}\right)\right\| + \frac{2T}{\Delta_1(n/T)}\left\|DX\left(\frac{n}{T}\right)\right\| \nonumber\\
&\quad+ \frac{2c_1(n/T)}{\pi (1-\cos(\Delta_1(n/T)/2))} \left\|X\left(\frac{n}{T}\right)\right\| + \frac{2\mathcal{G}_{T,4}(n_-/T)}{\Delta_0(n/T)}\left\|X\left(\frac{n}{T}\right)\right\|.
\end{align}
Our next step is the replacement of the bound of $X(s)$,
\begin{align}
\left\|Z\left(\frac{n}{T}\right)\right\| &\leq \frac{4T^2}{\Delta_0\left(n/T\right)} \left(\left\|\mathcal{F}_T\left(\frac{n}{T}\right) - I\right\| + \left\|DP_T\left(\frac{n}{T}\right)\right\|\left\|\mathcal{F}_T\left(\frac{n}{T}\right) \right\|\right)\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{\Delta_1(n_-/T)}\right) + \frac{2T}{\Delta_1(n/T)}\left\|DX\left(\frac{n}{T}\right)\right\| \nonumber\\
&\quad+ \frac{2Tc_1(n/T)}{\pi (1-\cos(\Delta_1(n/T)/2))} \mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right) + \frac{2T\mathcal{G}_{T,4}(n_-/T)}{\Delta_0(n/T)}\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right).
\end{align}
Now we use
\begin{equation}
\label{eq:comp_upper}
\left\|\mathcal{F}_T\left(\frac{n}{T}\right) - I\right\| + \left\|DP_T\left(\frac{n}{T}\right)\right\|\left\|\mathcal{F}_T\left(\frac{n}{T}\right) \right\|\leq \mathcal{D}_2\left(\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right),
\end{equation}
and the bound derived above for $DX(s)$ to yield
\begin{align}\label{eq:Zbnd}
\left\|Z\left(\frac{n}{T}\right)\right\| &\leq \frac{4T^2}{\Delta_0\left(n/T\right)} \mathcal{D}_2\left(\frac{2c_1(n/T)}{\Delta_1(n/T)}\right)\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{\Delta_1(n_-/T)}\right) + \frac{2\mathcal{G}_{T,3}(n_-/T)}{\Delta_1(n/T)} \nonumber\\
&\quad+ \frac{2Tc_1(n/T)}{\pi (1-\cos(\Delta_1(n/T)/2))} \mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right) + \frac{2T\mathcal{G}_{T,4}(n_-/T)}{\Delta_0(n/T)}\mathcal{D}_2\left(\frac{2c_1(n_-/T)}{T\Delta_1(n_-/T)}\right).
\end{align}
Finally, for the upper bound of $DY(s)$, first notice that $Y(s)=\Omega_T(s)$, so $D\Omega_T(s)=DY(s)$. Therefore, using \cref{lem:DOmega2} and our bound in \cref{eq:comp_upper} we obtain
\begin{equation}\label{eq:Ybnd}
\left\|DY\left(\frac{n}{T}\right) \right\| \leq \mathcal{D}_2\left(\frac{2c_1(n/T)}{T\Delta_1(n/T)}\right).
\end{equation}
\subsection{Upper bounds for the functions of \texorpdfstring{\cref{theoAdia}}{Theorem 15}}\label{sec:uppMainF}
Starting the assumption that $T \geq \max (4\hat{c}_1(s)/\check{\Delta}(s))$, from the upper bounds given in \cref{eq:upperB_D} and from the inequality $1-\cos(\theta/2) \geq \theta^2/\pi^2$ for all $0\leq \theta \leq \pi$, we have
\begin{align}\label{eq:GT1app}
\mathcal{G}_{T,1}(n_-/T) &= \frac{ c_1(n_-/T)^2+c_1(n_-/T)c_1(n/T)}{\pi (1-\cos(\Delta_2(n_-/T)/2))} + \frac{2c_2(n_-/T)}{\Delta_2(n_-/T)}\nonumber \\
& \leq \frac{ \pi c_1(n_-/T)^2 + \pi c_1(n_-/T)c_1(n/T)}{ \Delta_2(n_-/T)^2} + \frac{2c_2(n_-/T)}{\Delta_2(n_-/T)} \nonumber \\
& \leq \frac{2\pi\hat{c}_1(n/T)^2}{\check{\Delta}(n/T)^2} + \frac{2\hat{c}_2(n/T)}{\check{\Delta}(n/T)}.
\end{align}
Therefore
\begin{align}\label{eq:GT2app}
\mathcal{G}_{T,2}(n_-/T) & \leq \xi_3 \frac{2\hat{c}_1(n/T)}{T\check{\Delta}(n/T)} \left(\frac{2\pi\hat{c}_1(n/T)^2}{\check{\Delta}(n/T)^2} + \frac{2\hat{c}_2(n/T)}{\check{\Delta}(n/T)}\right) \nonumber \\
& = \frac{4\pi\xi_3 \hat{c}_1(n/T)^3}{T\check{\Delta}(n/T)^3} + \frac{4\xi_3 \hat{c}_1(n/T)\hat{c}_2(n/T)}{T\check{\Delta}(n/T)^2},
\end{align}
\begin{align}\label{eq:GT3app}
\mathcal{G}_{T,3}(n_-/T) &\leq \frac{3}{2}\left(\frac{4\pi\xi_3 \hat{c}_1(n/T)^3}{T\check{\Delta}(n/T)^3} + \frac{4\xi_3\hat{c}_1(n/T)\hat{c}_2(n/T)}{T\check{\Delta}(n/T)^2}\right) + \xi_1 \left(\frac{2\pi\hat{c}_1(n/T)^2}{\check{\Delta}(n/T)^2} + \frac{2\hat{c}_2(n/T)}{\check{\Delta}(n/T)} + \frac{8\hat{c}_1(n/T)^2}{\check{\Delta}(n/T)^2}\right) \nonumber \\
& \leq \frac{3}{2}\left(\frac{\pi\xi_3 \hat{c}_1(n/T)^2}{\check{\Delta}(n/T)^2} + \frac{\xi_3 \hat{c}_2(n/T)}{\check{\Delta}(n/T)}\right) + \xi_1\left(\frac{(2\pi+8)\hat{c}_1(n/T)^2}{\check{\Delta}(n/T)^2} + \frac{2\hat{c}_2(n/T)}{\check{\Delta}(n/T)}\right) \nonumber \\
& \leq \left(3\pi\xi_3/2 + (2\pi+8)\xi_1\right)\frac{\hat{c}_1(n/T)^2}{\check{\Delta}(n/T)^2} + \left(3\xi_3/2+2\xi_1\right)\frac{\hat{c}_2(n/T)}{\check{\Delta}(n/T)},
\end{align}
and
\begin{align}\label{eq:GT4app}
\mathcal{G}_{T,4}(n_-/T) \leq \left(3\pi\xi_3/2 + (2\pi+8)\xi_1\right)\frac{\hat{c}_1(n/T)^2}{T\check{\Delta}(n/T)^2} + \left(3\xi_3/2+2\xi_1\right)\frac{\hat{c}_2(n/T)}{T\check{\Delta}(n/T)} + \hat{c}_1(n/T).
\end{align}
These bounds are used in the body of the paper in \cref{eq:GT1body,eq:GT2body,eq:GT3body,eq:GT4body}.
Plugging all these bounds back to \cref{theoAdia} and using $1-\cos(\theta/2) \geq \theta^2/\pi^2$ again, we have
\begin{align}\label{eq:Uerrbndapp}
& \quad \|U_T(s) - U_T^A(s)\| \nonumber\\
&\leq \frac{8\xi_2 \hat{c}_1(0)}{T\check{\Delta}(0)^2}+ \frac{8\xi_2 \hat{c}_1(s)}{T\check{\Delta}(s)^2} + \frac{4\xi_2 \hat{c}_1(s)}{T\check{\Delta}(s)} + \sum_{n=1}^{sT-1}\frac{12}{\check{\Delta}(n/T)}\xi_2^2\left(\frac{2\hat{c}_1(n/T)}{T\check{\Delta}(n/T)}\right)^2 \nonumber\\
& \quad + \sum_{n=1}^{sT-1}\left(6\pi\xi_3 + (8\pi+32)\xi_1\right)\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} + \sum_{n=1}^{sT-1}\left(6\xi_3+8\xi_1\right)\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2} + \sum_{n=1}^{sT-1} \frac{4\pi \hat{c}_1(n/T)}{ T \check{\Delta}(n/T)^2} \xi_2 \frac{2\hat{c}_1(n/T)}{T\check{\Delta}
(n/T)} \nonumber \\
& \quad + \sum_{n=1}^{sT-1}\left(6\pi\xi_3 + (8\pi+32)\xi_1\right) \frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3}\left(\xi_2\frac{2\hat{c}_1(n/T)}{T\check{\Delta}
(n/T)}\right) + \sum_{n=1}^{sT-1}\left(6\xi_3 + 8\xi_1\right) \frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2}\left(\xi_2\frac{2\hat{c}_1(n/T)}{T\check{\Delta}
(n/T)}\right) \nonumber \\
& \quad + \sum_{n=1}^{sT-1} \frac{4\hat{c}_1(n/T)}{T\check{\Delta}(n/T)}\left(\xi_2\frac{2\hat{c}_1(n/T)}{T\check{\Delta}
(n/T)}\right) + \sum_{n=0}^{sT-1} \frac{24\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^2} + \sum_{n=0}^{sT-1}\frac{8\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^2} \nonumber \\
& \leq \frac{8\xi_2\hat{c}_1(0)}{T\check{\Delta}(0)^2}+ \frac{8\xi_2\hat{c}_1(s)}{T\check{\Delta}(s)^2} + \frac{4\xi_2\hat{c}_1(s)}{T\check{\Delta}(s)} + \sum_{n=1}^{sT-1}\frac{48\xi_2^2\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} \nonumber\\
& \quad + \sum_{n=1}^{sT-1}\left(6\pi\xi_3+(8\pi+32)\xi_1\right)\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} + \sum_{n=1}^{sT-1}(6\xi_3+8\xi_1)\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2}+ \sum_{n=1}^{sT-1} \frac{8\pi\xi_2 \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^3} \nonumber \\
& \quad + \sum_{n=1}^{sT-1}\left(12\pi\xi_2\xi_3+(16\pi+64)\xi_1\xi_2\right) \frac{\hat{c}_1(n/T)^3}{T^3\check{\Delta}(n/T)^4} + \sum_{n=1}^{sT-1} \left(12\xi_2\xi_3+16\xi_1\xi_2\right)\frac{\hat{c}_1(n/T)\hat{c}_2(n/T)}{T^3\check{\Delta}(n/T)^3} \nonumber \\
& \quad + \sum_{n=1}^{sT-1} \frac{8\xi_2 \hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^2} + \sum_{n=0}^{sT-1} \frac{24\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^2} + \sum_{n=0}^{sT-1}\frac{8\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^2} \nonumber \\
& \leq \frac{8\xi_2\hat{c}_1(0)}{T\check{\Delta}(0)^2}+ \frac{8\xi_2\hat{c}_1(s)}{T\check{\Delta}(s)^2} + \frac{4\xi_2\hat{c}_1(s)}{T\check{\Delta}(s)} + \sum_{n=1}^{sT-1} \left(48\xi_2^2 + 6\pi \xi_3 + (8\pi+32)\xi_1 + 8\pi \xi_2\right) \frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} \nonumber \\
& \quad + \sum_{n=1}^{sT-1}(6\xi_3+8\xi_1)\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2}+ \sum_{n=0}^{sT-1} \frac{(32+8\xi_2) \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^2} \nonumber \\
& \quad + \sum_{n=1}^{sT-1}\left(12\pi\xi_2\xi_3+(16\pi+64)\xi_1\xi_2\right) \frac{\hat{c}_1(n/T)^3}{T^3\check{\Delta}(n/T)^4} + \sum_{n=1}^{sT-1} \left(12\xi_2\xi_3+16\xi_1\xi_2\right)\frac{\hat{c}_1(n/T)\hat{c}_2(n/T)}{T^3\check{\Delta}(n/T)^3}.
\end{align}
This result is used in the body of the paper in \cref{eq:Uerrbndbody}.
\section{Block encoding of \texorpdfstring{$H(s)$}{H(s)}}\label{app:blockHs}
Here we describe how to perform the block encoding of $H(s)$ as given in \cref{eq:Hsencoding}.
We denote the unitary for the block encoding of $A$ as $U_A$, which acts on an ancilla denoted with subscript $a$ and the system such that
\begin{equation}
{}_a \! \bra{0} U_A \ket{0}_a = A.
\end{equation}
We also denote the unitary oracle for preparing $\ket{b}$ as $U_b$ such that
\begin{equation}
U_b \ket{0} = \ket{b}.
\end{equation}
As well as the ancilla system used for the block encoding of $A$, we use four ancilla qubits.
These ancilla qubits are used as follows.
\begin{enumerate}
\item The first selects between the blocks in $A(f)$.
\item The next is used for preparing the combination of $\sigma_z\otimes I$ and $\mathbf{A}$.
\item The third is used in implementing $Q_\mathbf{b}$.
\item The fourth selects between the blocks in $H(s)$.
\end{enumerate}
These three qubits will be denoted with subscripts $a_1$ to $a_4$.
First consider $A(f)$, which can be written as
\begin{equation}
A(f) = (1-f) \sigma^z_{a_1} \otimes I_N + f (\ketbra{0}{1}_{a_1}\otimes A + \ketbra{1}{0}_{a_1}\otimes A^\dagger).
\end{equation}
Note that the first operators in the tensor products here, $\sigma^z$ and $\ketbra{0}{1}$ or $\ketbra{1}{0}$, act upon the ancilla denoted $a_1$.
To block encode the operation using ancilla $a_2$, we can use the select operation
\begin{equation}
U_{A(f)} = \ketbra{0}{0}_{a_2} \otimes \sigma^z_{a_1} \otimes I_{N} \otimes I_a +
\ketbra{1}{1}_{a_2} \otimes (\ketbra{0}{1}_{a_1}\otimes U_A + \ketbra{1}{0}_{a_1}\otimes U_A^\dagger).
\end{equation}
Here we have included $I_a$ to indicate that the operation is acting as the identity on the ancilla system used for the block encoding of $A$.
Note that we require the ability to apply the oracle $U_A$ in a selected way, where we either perform $U_A$, $U_A^\dagger$ or the identity.
Next consider $Q_{\mathbf{b}}$, which is given by
\begin{equation}
Q_{\mathbf{b}} = I_{a_1} \otimes I_N - \ketbra{1}{1}_{a_1} \otimes \ketbra{b}{b}.
\end{equation}
Here we have used the ancilla $a_1$ to account for using $\mathbf{b}$ which is encoded as $\ket{1}_{a_1}\otimes \ket{b}$.
We can construct this projector using
\begin{equation}
(I_{a_1} \otimes U_b^\dagger) \left[ I_{a_1} \otimes I_N - \ketbra{1}{1}_{a_1} \otimes \ketbra{0}{0}_N \right] (I_{a_1} \otimes U_b),
\end{equation}
where we are using subscript $N$ on $\ketbra{0}{0}$ to indicate it is on the system.
We can block encode this projector using the ancilla $a_4$.
We simply need to create this ancilla in an equal superposition, and use the unitary operation
\begin{equation}
U_{Qb} = (I_{a_3} \otimes I_{a_1} \otimes U_b^\dagger) \left[ \ketbra{0}{0}_{a_3} \otimes I_{a_1} \otimes I_N
+\ketbra{1}{1}_{a_3} \otimes (2I_{a_1} \otimes I_N
- \ketbra{1}{1}_{a_1} \otimes \ketbra{0}{0}_N) \right] (I_{a_3} \otimes I_{a_1} \otimes U_b).
\end{equation}
That gives the projector as a linear combination of the identity and a reflection.
Finally we are prepared to describe the unitary to block encode $H(s)$, which can be written as
\begin{equation}
H(s) = \ketbra{0}{1}_{a_4} \otimes A(f(s)) Q_{\mathbf{b}} + \ketbra{1}{0}_{a_4} \otimes Q_{\mathbf{b}} A(f(s)).
\end{equation}
In order to select between $A(f(s)) Q_{\mathbf{b}}$ and $Q_{\mathbf{b}} A(f(s))$, we will apply $Q_{\mathbf{b}}$ in a controlled way before and after $A(f(s))$.
We will denote the controlled unitary for $Q_{\mathbf{b}}$, as controlled on 0 or 1, by $CU^0_{Qb}$ or $CU^1_{Qb}$, respectively.
We may make $Q_{\mathbf{b}}$ controlled simply by making the reflection $2I_{a_1} \otimes I_N
- \ketbra{1}{1}_{a_1} \otimes \ketbra{0}{0}_N$ controlled, and we do not need to make the oracle $U_b$ controlled.
We can therefore apply $CU^1_{Qb}$ as
\begin{align}
CU^1_{Qb} &= \ketbra{0}{0}_{a_4} \otimes I_{a_3} \otimes I_{a_1} \otimes I_N + \ketbra{1}{1}_{a_4} \otimes U_{Qb} \nonumber \\
&= (I_{a_3} \otimes I_{a_1} \otimes U_b^\dagger) \left[ I_{a_4} \otimes \ketbra{0}{0}_{a_3} \otimes I_{a_1} \otimes I_N
+\ketbra{0}{0}_{a_4} \otimes \ketbra{1}{1}_{a_3} \otimes I_{a_1} \otimes I_N \right. \nonumber \\ & \quad \left.
+\ketbra{1}{1}_{a_4} \otimes \ketbra{1}{1}_{a_3} \otimes (2I_{a_1} \otimes I_N
- \ketbra{1}{1}_{a_1} \otimes \ketbra{0}{0}_N)
\right] (I_{a_3} \otimes I_{a_1} \otimes U_b)
\end{align}
and similarly for $CU^0_{Qb}$.
We also need to perform the rotation $R(s)$ before or after these operations controlled on the ancilla $a_4$.
That is, we will perform at the beginning
\begin{equation}
CR^0(s) = \ketbra{0}{0}_{a_4} \otimes R(s)_{a_2} + \ketbra{1}{1}_{a_4} \otimes \mathcal{H}_{a_2},
\end{equation}
where $\mathcal{H}$ denotes the Hadamard operation.
Then at the end we perform the controlled operation
\begin{equation}
CR^1(s) = \ketbra{1}{1}_{a_4} \otimes R(s)_{a_2} + \ketbra{0}{0}_{a_4} \otimes \mathcal{H}_{a_2},
\end{equation}
We are finally ready to provide the complete sequence of operations to block encode $H(s)$.
In the following we will use the various operations defined above on subsets of the ancillas, with the convention that they act as the identity on any ancillas their action has not been described on.
\begin{enumerate}
\item We apply the Hadamard on $a_3$ to provide the linear combination needed for $Q_\mathbf{b}$.
\item Next apply $CU^1_{Qb}$ for controlled implementation of $Q_\mathbf{b}$ before $A(f(s))$.
\item Apply $CR^0(s)$ to provide the rotation on ancilla $a_2$.
\item Apply $U_{A(f)}$ for the block encoding of $A(f(s))$.
\item Apply $CR^1(s)$ to provide the symmetric form of the rotation on ancilla $a_2$.
\item Apply $CU^0_{Qb}$ for controlled implementation of $Q_\mathbf{b}$ after $A(f(s))$.
\item Apply the Hadamard on $a_3$ again.
\item Finally, apply $\sigma^x$ on $a_4$ to flip that bit.
\end{enumerate}
Now recall that we require the unitary operation in the block encoding to be self-inverse for the qubitisation.
To see that this sequence of operations is self-inverse, first note that each of the individual operations is self-inverse.
Second, note that $CU^0_{Qb}=\sigma^x_{a_4} CU^0_{Qb} \sigma^x_{a_4}$ and $CR^0(s)=\sigma^x_{a_4}CR^1(s)\sigma^x_{a_4}$.
That is, we may flip between controlling on 0 and 1 by applying the \textsc{not} gate to $a_4$.
Therefore we have the complete operation squared given by
\begin{align}
&\left[ \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,CR^1(s)\,U_{A(f)}\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3}\right]
\left[ \sigma^x_{a_4}\, \mathcal{H}_{a_3} \,CU^0_{Qb}\,CR^1(s)\,U_{A(f)}\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3} \right]\nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,CR^1(s)\,U_{A(f)}\,CR^0(s)\,CU^1_{Qb}\,
\sigma^x_{a_4}\, CU^0_{Qb}\,CR^1(s)\,U_{A(f)}\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,CR^1(s)\,U_{A(f)}\,CR^0(s)\,\sigma^x_{a_4}\,CU^0_{Qb}\,
CU^0_{Qb}\,CR^1(s)\,U_{A(f)}\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,CR^1(s)\,U_{A(f)}\,CR^0(s)\,\sigma^x_{a_4}\,CR^1(s)\,U_{A(f)}\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,CR^1(s)\,U_{A(f)}\,\sigma^x_{a_4}\,CR^1(s)\,CR^1(s)\,U_{A(f)}\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,CR^1(s)\,U_{A(f)}\,\sigma^x_{a_4}\,U_{A(f)}\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,CR^1(s)\,\sigma^x_{a_4}\,U_{A(f)}\,U_{A(f)}\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,CR^1(s)\,\sigma^x_{a_4}\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,\sigma^x_{a_4}\,CR^0(s)\,CR^0(s)\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, CU^0_{Qb}\,\sigma^x_{a_4}\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, \sigma^x_{a_4}\,CU^1_{Qb}\,CU^1_{Qb}\,\mathcal{H}_{a_3} \nonumber \\
&= \sigma^x_{a_4} \,\mathcal{H}_{a_3}\, \sigma^x_{a_4}\,\mathcal{H}_{a_3} \nonumber \\
&= I.
\end{align}
Here we have repeatedly commuted $\sigma^x_{a_4}$ through operators, and used the property that operators are self-inverse to cancel them.
This shows that our sequence of operations is self-inverse as required.
\section{Upper bounds of \texorpdfstring{\cref{cor:adia}}{Theorem 3} with \texorpdfstring{$p=3/2$}{p=3/2}}\label{ap:upperB_cor}
We split the proof of \cref{theo:p15} into three parts: the upper bounds for the three terms without sums, the sums with $\hat{c}_1$, and the summation term with $\hat{c}_2$.
Before we proceed with each calculation, first we note that in \cref{cor:adia} the three gaps are replaced by the minimum one, \cref{eq:fhat}, that is
\begin{equation}
\check{\Delta}(s) = \min_{s' \in \left\{s-1/T,s,s+1/T\right\} \cap [0,1] } \Delta(s').
\end{equation}
We have \cref{eq:Gaps} using the fact that the gap is monotonically decreasing, so then the fact that $f$ is monotonically increasing gives us
\begin{equation}
\label{eqn:qlsp_gap_check}
\check{\Delta}(s) = \begin{cases}
(1- f(s+3/T) + f(s+3/T)/\kappa), & 0 \leq s \leq 1-3/T, \\
1/\kappa, & s = 1-2/T, 1-1/T, 1. \\
\end{cases}
\end{equation}
Next, in \cref{cor:adia} we have the functions $\hat{c}_1(s)$ and $\hat{c}_2(s)$ as defined in \cref{eq:chat}.
Choices for the functions $c_1(s)$ and $c_2(s)$ are given in \cref{lem:DR}.
Using the monotonicity properties of the function $f$, we find
\begin{equation}
\label{eq:cic1corr}
\hat{c}_1(s) = \begin{cases}
2Tf(1/T), & s= 0, \\
2T(f(s)-f(s-1/T)), & 1/T \leq s \leq 1,
\end{cases}
\end{equation}
and
\begin{equation}
\label{eq:cic2corr}
\hat{c}_2(s) = 2\left(2|f'(s)|^2 + |f''(s)|\right).
\end{equation}
For $\hat{c}_1(s)$ we have used the fact that $f'(s)$ is monotonically decreasing, so a larger difference will be obtained for a smaller value of $s$.
For $\hat{c}_2(s)$ we have also used the fact that $|f''(s)|$ is monotonically decreasing, so again larger values will be obtained for smaller values of $s$.
The monotonicity properties of $f$ are easily checked by checking expressions for the derivatives; $f'(s)$ is positive, $f''(s)$ is negative and $f'''(s)$ is positive.
In the block encoding we need to account for how the gap in $H(s)$ is translated to the gap in the walk operators.
The solution state has eigenvalue $0$, which is translated to the eigenvalues $\pm 1$ for the walk operator.
The eigenvalues $\lambda$ of $H$ are generally translated to $\pm e^{\pm i\arcsin \lambda}$, which means the gap for the walk operator is increased to the arcsine of the gap of the Hamiltonian.
Since the arcsine can only increase the gap, the lower bounds on the gap for $H(s)$ also apply to the walk operator.
\subsection{Single components}\label{appsec:single}
Beginning with the first term from the bound in \cref{cor:adia}, using the expression for $\check{\Delta}(0)$ from \cref{eqn:qlsp_gap_check}, for $\hat{c}_1(0)$ from \cref{eq:cic1corr}, and $f(s)$ from \cref{eq:sched1}, we get
\begin{align}
\label{eq:firstupperB}
\frac{\hat{c}_1(0)}{T\check{\Delta}(0)^2}& = 2 \frac{f(1/T)}{(1-f(3/T)+f(3/T)/\kappa)^2}\nonumber\\
&=\frac{2}{ T^4}\frac{\kappa}{\sqrt{\kappa}+1}\frac{\left(3\sqrt{\kappa}-3+T\right)^4\left(\sqrt{\kappa}-1+2T\right)}{(\sqrt{\kappa}-1+T)^2}\nonumber\\
&=\frac{4}{ T}\frac{\kappa}{\sqrt{\kappa}+1} \frac{(1+2\alpha_1)^4 (1-\alpha_1/2)}{(1-\alpha_1)^3} \nonumber \\
&=\frac{4}{ T}\frac{\kappa}{\sqrt{\kappa}+1} \left[1+\mathcal{O}(\alpha_1)\right] \nonumber \\
&= \frac{4\sqrt{\kappa}}{T} + \mathcal{O}\left( \frac{\kappa}{T^2} \right),
\end{align}
where
\begin{equation}
\alpha_n:=\frac{\sqrt\kappa-1}{T+n(\sqrt\kappa-1)},
\end{equation}
so $\alpha_n=\mathcal{O}(\sqrt\kappa/T)$,
and we have used $T>\kappa$.
This result is given in \cref{eq:single_c(0)} of the body.
We next show \cref{eq:single_c(1),eq:single_c(1)_2}.
This time we use $\hat{c}_1(s)$ and $\check{\Delta}(s)$ for $s=1$; by \cref{eq:cic1corr} we get $\hat{c}_1(1)=2(1-f(1-1/T))$ and from \cref{eqn:qlsp_gap_check} we have $\check{\Delta}(1)=1/\kappa$.
Therefore
\begin{align}
\frac{\hat{c}_1(1)}{T\check{\Delta}(1)^2}& = 2\kappa^2 (1-f(1-1/T))\nonumber\\
& = 2\kappa^2 \left[1+\frac{\kappa}{1-\kappa}\left(1-\frac{1}{(1+(\sqrt{\kappa} -1)(1-1/T))^2}\right)\right].
\end{align}
Now we simplify the terms inside the square brackets to give
\begin{align}
\label{eq:bracket}
1+\frac{\kappa}{1-\kappa}\left[1-\frac{T^2}{(T+(\sqrt{\kappa} -1)(T-1))^2}\right] &= \frac{(1-\kappa)\left(1+\sqrt{\kappa}(T-1)\right)^2+\kappa\left(1+\sqrt{\kappa}(T-1)\right)^2 - T^2\kappa}{\left(1 - \kappa \right)\left(1+\sqrt{\kappa}(T-1)\right)^2}\nonumber\\
&= \frac{\left(1+\sqrt{\kappa}(T-1)\right)^2 - T^2\kappa}{\left(1 - \kappa \right)\left(1+\sqrt{\kappa}(T-1)\right)^2}\nonumber\\
&=\frac{\sqrt{\kappa}( 2 T-1) +1}{(\sqrt{\kappa}+1)(1+\sqrt{\kappa}(T-1))^2} \nonumber \\
&=\frac{2}{T(\kappa+\sqrt{\kappa})}
\frac{1-\beta/2}{(1-\beta)^2} \nonumber \\
&=\frac{2}{T(\kappa+\sqrt{\kappa})} \left[1+\mathcal{O}(\beta)\right] \nonumber \\
&=\frac{2}{\kappa T} + \mathcal{O}\left( \frac{1}{\kappa T^2} \right),
\end{align}
with
\begin{equation}
\beta=\frac{1-1/\sqrt\kappa} T,
\end{equation}
so $\beta=\mathcal{O}(1/T)$.
Therefore, we can conclude
\begin{equation}
\label{eq:SecondupperB}
\frac{\hat{c}_1(1)}{ T\check{\Delta}(1)^2}
=\frac{4\kappa}{T} + \mathcal{O}\left( \frac{\kappa}{T^2} \right).
\end{equation}
This is the result given in \cref{eq:single_c(1)}.
For the other upper bound, we have $\check{\Delta}(1)$ instead of $\check{\Delta}(1)^2$, so get for the upper bound shown in \cref{eq:single_c(1)_2}
\begin{equation}
\label{eq:thirdupperB}
\frac{\hat{c}_1(1)}{T\check{\Delta}(1)}
=\frac{4}{T} + \mathcal{O}\left( \frac{1}{T^2} \right).
\end{equation}
\subsection{\texorpdfstring{$c_1(s)$}{c1(s)} summations}
\label{appsec:summc1}
We start by considering the sum of $\hat{c}_1(s)^2/(T^2\check{\Delta}(s)^3)$ for $1/T\leq s\leq 1-3/T$.
In this range we get
\begin{align}
\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} &= 4 \frac{(f(n/T)-f((n-1)/T))^2}{(1-f((n+3)/T)+f((n+3)/T)/\kappa)^3} \nonumber \\
&= \frac{16\kappa^2}{(\sqrt\kappa + 1)^2T^2}\frac{\left[(3 + n)(\sqrt{\kappa}-1)+T\right]^6\left[(n - 1/2)(\sqrt{\kappa}-1)+T\right]^2}{\left[n(\sqrt{\kappa}-1)+T\right]^4\left[(n-1)(\sqrt{\kappa}-1)+T\right]^4}\nonumber\\
&= \frac{16\kappa^2}{(\sqrt\kappa + 1)^2T^2}
\frac{(1+3\alpha_n)^6(1-\alpha_n/2)^2}{(1-\alpha_n)^4}
\nonumber\\
&= \frac{16\kappa^2}{(\sqrt\kappa + 1)^2T^2}
\left[1+\mathcal{O}(\alpha_n)\right] \nonumber \\
&= \frac{16 \kappa}{T^2}+ \mathcal{O} \left( \frac{\kappa^{3/2}}{T^3} \right).
\end{align}
Now for the last two elements of the sum we have
\begin{align}
\label{eq:upper-1/t}
\frac{\hat{c}_1(1-2/T)^2}{T^2\check{\Delta}(1-2/T)^3} &= 4 \kappa^3(f(1-2/T) - f(1-3/T) )^2\nonumber \\
&=
\frac{16 \kappa^2}{
T^2 (1 + \sqrt\kappa)^2} \frac{[1 - 5/(2 T) + 5/(
2 T \sqrt\kappa)]^2}{[1 + 6/T^2 - 5/T + 6/(T^2 \kappa) - 12/(
T^2 \sqrt\kappa) + 5/(T \sqrt\kappa)]^4}\nonumber \\
&=\frac{16 \kappa^2}{
T^2 (1 + \sqrt\kappa)^2} \left[ 1+\mathcal{O}\left( \frac 1T \right) \right] \nonumber \\
&=\frac{16 \kappa}{T^2}+ \mathcal{O} \left( \frac{\kappa}{T^3} \right),
\end{align}
and
\begin{align}
\label{eq:upper-2/t}
\frac{\hat{c}_1(1-1/T)^2}{T^2\check{\Delta}(1-1/T)^3} &= 4\kappa^3(f(1-1/T) - f(1-2/T) )^2 \nonumber \\
&=
\frac{16 \kappa^2}{
T^2 (1 + \sqrt\kappa)^2} \frac{[1 - 3/(2 T) + 3/(
2 T \sqrt\kappa)]^2}{[1 + 2/T^2 - 3/T + 2/(T^2 \kappa) - 4/(
T^2 \sqrt\kappa) + 3/(T \sqrt\kappa)]^4}\nonumber \\
&=\frac{16 \kappa^2}{
T^2 (1 + \sqrt\kappa)^2} \left[ 1+\mathcal{O}\left( \frac 1T \right) \right] \nonumber \\
&=\frac{16 \kappa}{T^2}+ \mathcal{O} \left( \frac{\kappa}{T^3} \right).
\end{align}
Therefore, for all $n$ in the sum we have an upper bound of $16\kappa/T^2$ up to leading order.
The total upper bound for the sum of $\hat{c}_1(n/T)^2/(T^2\check{\Delta}(n/T)^3)$ from from $n=1$ to $T-1$ is therefore
\begin{equation}
\label{eq:uppersum1}
\sum_{n=1}^{T-1} \hat{c}_1(n/T)^2/(T^2\check{\Delta}(n/T)^3) = \frac{16 \kappa}{T}+ \mathcal{O} \left( \frac{\kappa^{3/2}}{T^2} \right),
\end{equation}
which is given in \cref{eq:sumc1}.
Next we show the upper bound for the sum of the elements $\hat{c}_1(s)^2/(T^2\check{\Delta}(s)^2)$, which is given in \cref{eq:sumc1_2} above. When $1/T\leq s\leq 1-3/T$ we have
\begin{align}
\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^2}&= 4 \frac{(f(n/T)-f((n-1)/T))^2}{(1-f((n+3)/T)+f((n+3)/T)/\kappa)^2}\nonumber\\
&= \frac{4\kappa^2}{(\sqrt{\kappa} + 1)^2}\frac{\left[(3+n)(\sqrt{\kappa}-1)+T\right]^4 \left[ \left(2n - 1\right)\left(\sqrt{\kappa} - 1\right) + 2T\right]^2 }{\left[n(\sqrt{\kappa}-1)+T\right]^4\left[(n-1)(\sqrt{\kappa}-1)+T\right]^4}\nonumber\\
&= \frac{16\kappa^2}{[T+n(\sqrt\kappa-1)]^2(\sqrt{\kappa} + 1)^2} \frac{(1-\alpha_n/2)^2(1+3\alpha_n)^4}{(1-\alpha_n)^4}\nonumber \\
&= \frac{16\kappa^2}{[T+n(\sqrt\kappa-1)]^2(\sqrt{\kappa} + 1)^2} \left[ 1+ \mathcal{O}\left(\alpha_n \right)\right]\nonumber \\
&\le \frac{16}{T^2}+\mathcal{O}\left(\frac {\sqrt\kappa}{T^3} \right).
\end{align}
Because the sum starts from $n=0$ we need the following upper bound
\begin{align}
\frac{\hat{c}_1(0)^2}{T^2\check{\Delta}(0)^2}&=4\frac{f(1/T)^2}{(1-f(3/T)+f(3/T)/\kappa)^2}\nonumber\\
& = \frac{4\kappa^2}{(\sqrt{\kappa} + 1)^2}\frac{\left(3(\sqrt{\kappa}-1) + T\right)^4\left(\sqrt{\kappa} - 1 + 2T\right)^2}{T^4\left(\sqrt{\kappa}-1+T\right)^4}\nonumber\\
& = \frac{16\kappa^2}{(\sqrt\kappa+1)^2T^2} \frac{(1+a_0/2)^2(1+3a_0)^4}{(1+a_0)^4} \nonumber \\
& = \frac{16\kappa^2}{(\sqrt\kappa+1)^2T^2} \left[ 1+\mathcal{O}(a_0)\right] \nonumber \\
&= \frac{16\kappa}{T^2} + \mathcal{O} \left( \frac {\kappa^{3/2}}{T^3} \right).
\end{align}
We also have to upper bound the cases where $s=1-1/T$ and $s=1-2/T$. This upper bound is the same as we had in~\cref{eq:upper-1/t,eq:upper-2/t}, but now with $1/\kappa^2$ in the denominator rather than $1/\kappa^3$, so we get
\begin{equation}
\frac{\hat{c}_1(1-2/T)^2}{T^2\check{\Delta}(1-2/T)^2} = 4\kappa^2(f(1-2/T) - f(1-3/T) )^2 = \frac{16}{T^2} + \mathcal{O} \left( \frac {1}{T^3} \right),
\end{equation}
and
\begin{equation}
\frac{\hat{c}_1(1-1/T)^2}{T^2\check{\Delta}(1-1/T)^2} = 4\kappa^2(f(1-1/T) - f(1-2/T) )^2= \frac{16}{T^2} + \mathcal{O} \left( \frac {1}{T^3} \right).
\end{equation}
There are $T$ terms in the sum, and each is upper bounded by $16/T^2$ to leading order except that at $n=0$.
We therefore get
\begin{equation}
\sum_{n=0}^{T-1}\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^2}
\leq \frac{16}{T} + \frac{16\kappa}{T^2} + \mathcal{O} \left( \frac {\sqrt\kappa}{T^2} \right)
= \frac{16}{T} +\mathcal{O} \left( \frac {\kappa}{T^2} \right).
\end{equation}
This is the result given as \cref{eq:sumc1_2} above.
\subsection{\texorpdfstring{$c_2(s)$}{c2(s)} summation}\label{appsec:sumc2}
Next we show the upper bound given in \cref{eq:sumc2}.
Using \cref{eq:cic2corr,eqn:qlsp_gap_check} for $1\leq s\leq 1 - 3/T$ we have
\begin{align}
\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2}&= \frac{1}{ T^2} \frac{4 |f'(n/T)|^2 + |f''(n/T)|}{(1-f((n+3)/T)+f((n+3)/T)/\kappa)^2}\nonumber\\
&= \frac{2\kappa}{T^2(\sqrt\kappa+1)^2} (1+3a_n)^4((3+8\gamma^2)\kappa-3) \nonumber \\
&\leq \frac{22\kappa}{T^2(\sqrt\kappa+1)^2}(1+3a_n)^4\nonumber \\
&= \frac{22\kappa}{T^2} + \mathcal{O}\left(\frac{\kappa^{3/2}}{T^3}\right),
\end{align}
where
\begin{equation}
\gamma=\frac T{T+n(\sqrt\kappa-1)}<1.
\end{equation}
We need to separately consider the case $s=1-1/T$, which gives
\begin{align}
\frac{\hat{c}_2(1-1/T)}{T^2\check{\Delta}(1-1/T)^2}&= \frac{\kappa^2}{ T^2} \left(4|f'(1-1/T)|^2 + |f''(1-1/T)|\right)\nonumber\\
&= \frac{2T^2\kappa^3}{(\sqrt\kappa+1)^2(\sqrt\kappa(T-1)+1)^4} (3\kappa + 5 + 16\delta + 8\delta^2) \nonumber \\
&= \frac {6\kappa}{T^2} + \mathcal{O}\left( \frac{1}{T^2}\right),
\end{align}
where
\begin{equation}
\delta = \frac{\sqrt\kappa-1}{\sqrt\kappa(T-1)+1} = \mathcal{O}(1/T).
\end{equation}
Similarly, we obtain
\begin{align}
\frac{\hat{c}_2(1-2/T)}{T^2\check{\Delta}(1-2/T)^2}&= \frac{\kappa^2}{ T^2} \left(4|f'(1-2/T)|^2 + |f''(1-2/T)|\right)\nonumber\\
&= \frac{2T^2\kappa^3}{(\sqrt\kappa+1)^2(\sqrt\kappa(T-2)+2)^4} (3\kappa + 5 + 32\delta + 32\delta^2) \nonumber \\
&= \frac {6\kappa}{T^2} + \mathcal{O}\left( \frac{1}{T^2}\right),
\end{align}
where this time
\begin{equation}
\delta = \frac{\sqrt\kappa-1}{\sqrt\kappa(T-2)+2} = \mathcal{O}(1/T).
\end{equation}
Finally, since there are $T-1$ terms in the sum, and each is upper bounded by $22\kappa/T^2$ to leading order, we get
\begin{equation}
\sum_{n=1}^{T-1} \frac{ \hat{c}_2(n/T)}{ T^2 \check{\Delta}(n/T)^2} \leq \frac {22\kappa}{T} + \mathcal{O}\left( \frac{\kappa^{3/2}}{T^2}\right).
\end{equation}
This is the bound given in \cref{eq:sumc2}.
\section{Phase factors in the adiabatic evolution}
\label{app:phasefactor}
One would normally consider the eigenspace of interest in a single group for the adiabatic theorem.
In contrast, here the eigenspace of interest is separated in two parts, corresponding to $\pm 1$, and the remaining eigenspace is separated by a gap in two parts in the upper and lower halves of the complex plane.
In order to address this, one can instead consider just the eigenvalue $1$ as the eigenspace of interest, which is then in a single group.
Then the discrete adiabatic theorem can be applied unchanged to show that the state is correctly mapped to the final eigenstate.
Similarly, one can just consider the adiabatic theorem with $-1$.
Since using the adiabatic theorem separately on each eigenstate shows that it properly evolves to the final state, the superposition of the two eigenstates must also.
To be more specific, as discussed in Eqs.\ (10) to (13) of \cite{BerryNPJ18}, the eigenvectors of the walk operator are of the form (correcting a missing $i$ in the reference)
\begin{equation}\label{eq:qubeig}
\frac{1}{\sqrt 2} \left( \ket{0}_a \ket{k}_s \pm i\ket{0k^\perp}_{as} \right),
\end{equation}
where $\ket{0}_a$ is the zero state on the ancilla, $\ket{k}_s$ is the eigenstate of $H$ of energy $E_k$ on the system, and $\ket{0k^\perp}_{as}$ is a state orthogonal to $\ket{0}_a$ on the ancilla.
In our case, the target eigenvalue of the Hamiltonian is $E_k=0$, which yields eigenvalues of $\pm 1$ of the walk operator with these two eigenstates.
When we have a positive superposition of the two eigenstates, then the resulting state is the solution given by $\ket{0}_a \ket{k}_s$.
In contrast, if we have a negative superposition of the two eigenstates, then the result is the non-solution state $\ket{0k^\perp}_{as}$.
In the adiabatic evolution, we start with the positive superposition, and we must maintain that positive superposition at the end in order to obtain the solution.
Therefore, we should show that there is no phase factor introduced by the adiabatic evolution.
To show this, it is again sufficient to consider the evolution of each eigenvalue on its own.
To obtain the phase factor, it is sufficient to consider the exact adiabatic evolution given by the adiabatic walk operators
\begin{align}
W_T^A(s) &= V_T(s) W_T(s) \nonumber \\
&= v_T(s',s)^{-1} S_T(s',s) W_T(s) \nonumber \\
&= [S_T(s',s) S_T^\dagger(s',s)]^{-1/2} S_T(s',s) W_T(s) \nonumber \\
&= [P_T(s')P_T(s)P_T(s')+ Q_T(s')Q_T(s)Q_T(s')]^{-1/2} [P_T(s')P_T(s)+ Q_T(s')Q_T(s)] W_T(s).
\end{align}
where $s'=s+1/T$. (We are swapping the $s'$ and $s$ from the way $S_T$ and $v_T$ were given originally.)
For the case we are interested in, there may be multiple states within the spectrum of interest, but they are orthogonal.
More specifically, there is the solution state (ground state of the Hamiltonian)
\begin{equation}
\begin{pmatrix}
{A}(f(s))^{-1} \bm{b} \\ 0
\end{pmatrix},
\end{equation}
as well as a non-solution state of the form
\begin{equation}
\begin{pmatrix}
0 \\ \bm{b}
\end{pmatrix}.
\end{equation}
In that case the product of projectors is of the form
\begin{equation}
P_T(s')P_T(s) = \sum_{j,j'} \ket{\lambda_j(s')}\braket{\lambda_j(s')}{\lambda_{j'}(s)}\bra{\lambda_{j'}(s)} =
\sum_{j} \braket{\lambda_j(s')}{\lambda_{j}(s)} \ket{\lambda_j(s')}\bra{\lambda_{j}(s)} ,
\end{equation}
and similarly
\begin{equation}
P_T(s')P_T(s)P_T(s') =
\sum_{j} |\braket{\lambda_j(s')}{\lambda_{j}(s)}|^2 \ket{\lambda_j(s')}\bra{\lambda_{j}(s)} ,
\end{equation}
We use $\ket{\lambda_j(s)}$ to indicate eigenstates, including for degenerate eigenvalues.
What this means is that we cannot flip between orthogonal eigenstates in the spectrum of interest during the (exact) adiabatic evolution, and we do not have the solution state leaking into the non-solution state.
For the eigenstate $\ket{\lambda_j(s)}$ in the spectrum of interest for $W_T(s)$, we have
\begin{align}
&[P_T(s')P_T(s)P_T(s')+ Q_T(s')Q_T(s)Q_T(s')]^{-1/2} [P_T(s')P_T(s)+ Q_T(s')Q_T(s)] W_T(s) \ket{\lambda_j(s)} \nonumber \\
&=\lambda_j(s) [P_T(s')P_T(s)P_T(s')+ Q_T(s')Q_T(s)Q_T(s')]^{-1/2} [P_T(s')P_T(s)+ Q_T(s')Q_T(s)] \ket{\lambda_j(s)} \nonumber \\
&=\lambda_j(s)\braket{\lambda_j(s')}{\lambda_{j}(s)} [P_T(s')P_T(s)P_T(s')+ Q_T(s')Q_T(s)Q_T(s')]^{-1/2} \ket{\lambda_j(s')} \nonumber \\
&=\lambda_j(s)\braket{\lambda_j(s')}{\lambda_{j}(s)} [|\braket{\lambda_j(s')}{\lambda_{j}(s)}|^2\ket{\lambda_j(s')}\bra{\lambda_{j}(s)}]^{-1/2} \ket{\lambda_j(s')} \nonumber \\
&= \lambda_j(s)\frac{\braket{\lambda_j(s')}{\lambda_{j}(s)}}{|\braket{\lambda_j(s')}{\lambda_{j}(s)}|}\ket{\lambda_j(s')}.
\end{align}
In the second line applying $W_T(s)$ gives the eigenvalue.
If this eigenstate is $\ket{\lambda_j(s)}$, then applying $P_T(s')P_T(s)+Q_T(s')Q_T(s)$ gives the updated state $\ket{\lambda_j(s')}$ times $\braket{\lambda_j(s')}{\lambda_{j}(s)}$.
Then applying $v_T(s',s)^{-1}$ cancels the magnitude of $\braket{\lambda_j(s')}{\lambda_{j}(s)}$, and we only have its phase.
For our application the eigenvalues of $W_T(s)$ are $\pm 1$, but provided the total number of steps of the walk is even then this sign flip cancels out.
Ideally we would show that $\braket{\lambda_j(s')}{\lambda_{j}(s)}$ is real in order to show that there are no spurious phase factors.
However, it is sufficient to just show that any phase factor from $\braket{\lambda_j(s')}{\lambda_{j}(s)}$ is the same between the $\pm 1$ eigenvectors of $W_T(s)$.
The eigenstates of $W_T(s)$ are given as in \cref{eq:qubeig}.
In order to describe the inner products between the eigenstates at successive time steps, let us use $k_1$ and $k_2$.
Then the inner product of eigenstates at successive steps is
\begin{equation}
\frac 12 \left( {}_a\!\bra{0}\, {}_s\!\bra{k_1} \mp i \, {}_{as}\!\bra{0k_1^\perp}\right)\left( \ket{0}_a \ket{k_2}_s \pm i\ket{0k_2^\perp}_{as} \right) =
\frac 12 \left( \, {}_s\!\braket{k_1}{k_2}_s +\, {}_{as}\! \braket{0k_1^\perp}{0k_2^\perp}_{as} \right).
\end{equation}
Here we have used $\pm$ to indicate that we are using $+1$ for both steps or $-1$ for both steps.
The crucial result here is that the inner product does \emph{not} depend on whether we were considering the $+1$ or $-1$ eigenstates.
This means that there may be a phase factor, but it will be the \emph{same} between the $\pm 1$ eigenstates.
The one caveat is that we need to use an even number of steps to avoid a $-1$ factor, but it is always possible to slightly adjust the schedule so that there is an even number of steps, and that does not change the asymptotic scaling of the number of steps needed.
The net result of this is that the adiabatic walk with the qubitised walk operator still works, despite there being separated eigenvalues at $\pm 1$.
Here we have not needed to use any special properties of the Hamiltonian other than that the eigenvalue of interest (for the Hamiltonian) is $0$, so this result may be used for applications other than solving linear equations.
If there were a nonzero eigenvalue that was \emph{known}, then it would be possible to add a multiple of the identity to rezero that eigenvalue, and the above method would again work.
\section{Upper bounds of \texorpdfstring{\cref{thm:qlsp_general_p}}{Theorem 18} with \texorpdfstring{$1<p<2$}{1<p<2}}\label{app:qlsp_general_p}
In order to show the linear dependence in $T$ on $\kappa$, we use \cref{cor:adia} and calculate the scaling of each term.
There are two main difficulties: estimates of finite differences of the walk operator and different discrete time points used in \cref{cor:adia}.
To overcome the first difficulty, we establish a connection between the discrete finite difference coefficients $c_k(s)$ and the corresponding continuous derivatives of the schedule function.
Then, according to the definition of the schedule function \cref{eq:gapCon}, this can be directly related to the spectrum gap and cancel with the denominators in the error bound.
For the second difficulty, we use continuity and monotonicity of the spectrum gap in the linear system problem to unify the time points with sacrifice of larger preconstants.
We first reformulate the coefficients $c_1$ and $c_2$, which have already been done in \cref{lem:DR}.
Here, we will use a slightly different version with continuous time values, that
\begin{equation}
\label{eq:c1}
c_1(s) = 2\max_{\tau\in[s,s+1/T]\cap[0,1]}|f'(\tau)|,
\end{equation}
and
\begin{equation}
c_2(s) = 2\max_{\tau\in[s,s+2/T]\cap[0,1]}(2|f'(\tau)|^2+|f''(\tau)|).
\end{equation}
Notice that the choices of $c_1$ and $c_2$ here are even larger than those in \cref{lem:DR}.
Then we can use the definition of the schedule function to establish the connection between $c_k(s)$ and the spectrum gap.
\begin{lemma}\label{lem:qlsp_c_bound}
Consider solving linear system problem using discrete adiabatic evolution with schedule function defined in \cref{eq:gapCon}.
Then the walk operators satisfy
\begin{enumerate}
\item for any $0 \leq s \leq 1-1/T$, we have
\begin{equation}
c_1(s) = 2d_p\Delta_0(s)^{p},
\end{equation}
\item for any $0 \leq s \leq 1-2/T$, we have
\begin{equation}
c_2(s) = 4 d_p^2\Delta_0(s)^{2p} + 2 d_p^2 p (1-1/\kappa) \Delta_0(s)^{2p-1}.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
According to \cref{lem:DR}, we only need to compute the derivatives of the schedule function.
The first order derivative directly comes from the definition of the schedule function that
\begin{equation}
f'(\tau) = d_p \Delta_0(s)^p.
\end{equation}
For the second order derivative, we have
\begin{align}
f''(\tau) &= \frac{d}{d\tau} \left(d_p \left(1-f(\tau)+f(\tau)/\kappa\right)^p\right) \nonumber \\
& = d_p p\left(1-f(\tau)+f(\tau)/\kappa\right)^{p-1} (-1+1/\kappa)f'(\tau) \nonumber \\
&= d_p^2 p (-1+1/\kappa) \Delta_0(s)^{2p-1}.
\end{align}
The proof is completed using the monotonicity of $\Delta_0$.
\end{proof}
In the error estimate in \cref{cor:adia}, we encounter taking maximum or minimum of several consequent time steps, which poses technical difficulty in calculating the scaling of the error.
In the following lemma, we show how to resolve the different time point issue.
\begin{lemma}\label{lem:qlsp_time_unify}
Let $\Delta_0(s)$ denote the spectrum gap of the time-dependent Hamiltonian used in solving linear system problem.
Assume $T \geq 16(\sqrt{2})^p \left(\frac{\kappa^{p-1}-1}{p-1}\right) = \mathcal{O}(\kappa^{p-1})$.
Then for any $s \leq s' \leq s+4/T$, we have
\begin{equation}
\Delta_0(s) \leq \frac{4}{3}\Delta_0(s').
\end{equation}
\end{lemma}
\begin{proof}
For simplicity we only consider the Hermitian positive definite case, since the gap in the general non-Hermitian case only differs from the positive definite case by a multiplication factor of $\sqrt{2}$.
We define $\Delta_{\text{linear}}(y) = 1-y+y/\kappa$.
Then $\Delta_0(s) = \Delta_{\text{linear}}(f(s))$.
Since $\Delta(s)$ is a monotonically decreasing function, it suffices to prove $\Delta_0(s)/\Delta_0(s+4/T) \leq 4/3$.
We first compute the derivative of the gap,
\begin{align}
\Delta_0'(s) = \frac{d}{ds} \left(\Delta_{\text{linear}}(f(s))\right) = \Delta_{\text{linear}}'(f(s))f'(s) = (-1+1/\kappa) d_p \Delta_0(s)^p.
\end{align}
Then for any $0\leq s \leq 1-4/T$,
\begin{equation}
|\Delta_0(s) - \Delta_0(s+4/T)| \leq \frac{4}{T} \max_{s'\in[s,s+4/T]} |\Delta_0'(s')| = \frac{4}{T}(1-1/\kappa) d_p\Delta_0(s)^p,
\end{equation}
and thus
\begin{align}
\frac{\Delta_0(s)}{\Delta_0(s+4/T)} &= 1 + \frac{\Delta_0(s)-\Delta_0(s+4/T)}{\Delta_0(s+4/T)} \nonumber \\
& \leq 1 + \frac{4(1-1/\kappa)d_p \Delta_0(s)^{p-1}}{T}\frac{\Delta_0(s)}{\Delta_0(s+4/T)} \nonumber \\
& \leq 1 + \frac{4(1-1/\kappa)d_p }{T}\frac{\Delta_0(s)}{\Delta_0(s+4/T)}.
\end{align}
It has been computed in~\cite{an2019quantum} that $d_p = \frac{2^{p/2}}{p-1}\frac{\kappa}{\kappa-1}(\kappa^{p-1}-1)$.
Together with the assumption that $T \geq 16(\sqrt{2})^p \left(\frac{\kappa^{p-1}-1}{p-1}\right)$, we have
\begin{equation}
\frac{4(1-1/\kappa)d_p }{T} = \frac{2^{p/2+2}}{T(p-1)}(\kappa^{p-1}-1) \leq \frac{1}{4},
\end{equation}
and thus
\begin{equation}
\frac{\Delta_0(s)}{\Delta_0(s+4/T)} \leq 1 + \frac{1}{4} \frac{\Delta_0(s)}{\Delta_0(s+4/T)},
\end{equation}
which implies $\Delta_0(s)/\Delta_0(s+4/T) \leq 4/3$.
\end{proof}
Now we are ready to prove \cref{thm:qlsp_general_p}, the complexity estimate of using discrete adiabatic evolution to solve linear system problems.
\begin{proof}[Proof of \cref{thm:qlsp_general_p}]
Without loss of generality, here we only prove the scenario with Hermitian positive definite matrix $A$.
This is because the spectrum gap in the general non-Hermitian case only differs by a multiplication factor of $\sqrt{2}$, which can be absorbed by changing the concrete definition of the constant factor $C_p$.
First notice that
\begin{equation}
4\frac{c_1(s)}{\Delta_1(s)} = \frac{8d_{p} \Delta_0(s)^{p}}{\Delta_0(s+1/T)} \leq \frac{32d_p\Delta_0(s)^{p}}{3\Delta_0(s)} \leq \frac{32d_p}{3}.
\end{equation}
Therefore the assumption that $T \geq 32d_p/3$ ensures that the assumption in \cref{cor:adia} satisfies.
Now we apply \cref{cor:adia} to bound the error.
To simplify the computation, we now unify the time in the hat and check notations by applying \cref{lem:qlsp_time_unify}.
More precisely,
\begin{equation}
\hat{c}_1(s) = \max_{s'\in\{s-1/T,s,s+1/T\}\cap [0,1-1/T]} c_1(s') = \begin{cases}
2d_p \Delta_0(0)^{p}, & s = 0, \\
2d_p \Delta_0(s-1/T)^{p}, & 1/T \leq s \leq 1.
\end{cases}
\end{equation}
Applying \cref{lem:qlsp_time_unify} to change all the discrete time to $s$, we have for all $0 \leq s \leq 1$,
\begin{equation}\label{eqn:qlsp_c1_hat_proof}
\hat{c}_1(s) \leq \frac{2^{2p+1}}{3^p}d_p \Delta_0(s)^{p}.
\end{equation}
Similarly,
\begin{equation}
\hat{c}_2(s) = \begin{cases}
4 d_p^2\Delta_0(0)^{2p} + 2p d_p^2 (1-1/\kappa) \Delta_0(0)^{2p-1}, & s = 0\\
4 d_p^2\Delta_0(s-1/T)^{2p} + 2p d_p^2 (1-1/\kappa) \Delta_0(s-1/T)^{2p-1}, & 1/T \leq s \leq 1-1/T,
\end{cases}
\end{equation}
and for all $0 \leq s \leq 1-1/T$,
\begin{equation}\label{eqn:qlsp_c2_hat_proof}
\hat{c}_2(s) \leq \frac{2^{4p+2}}{3^{2p}}d_p^2\Delta_0(s)^{2p} + \frac{2^{4p-1}}{3^{2p-1}} p d_p^2 (1-1/\kappa) \Delta_0(s)^{2p-1}.
\end{equation}
For the spectrum gap, by \cref{eqn:qlsp_gap_check} and \cref{lem:qlsp_time_unify}, we have
\begin{equation}\label{eqn:qlsp_gap_hat_proof}
\check{\Delta}(s) \geq \frac{3}{4} \Delta_0(s).
\end{equation}
Combining \cref{eqn:qlsp_c1_hat_proof,eqn:qlsp_c2_hat_proof,eqn:qlsp_gap_hat_proof} and the fact that
\begin{equation}\label{eqn:qlsp_bound_dp}
d_p = \frac{2^{p/2}}{p-1}\frac{\kappa}{\kappa-1} (\kappa^{p-1}-1) \leq \frac{2^{1+p/2}}{p-1} \kappa^{p-1},
\end{equation}
we are now ready to bound each term in the error bound in \cref{cor:adia}.
The first three terms (i.e. boundary terms) in \cref{cor:adia} can be bounded as follows:
\begin{equation}\label{eqn:qlsp_term1}
\frac{\hat{c}_1(0)}{T\check{\Delta}(0)^2} \leq \frac{2^{2p+1}}{3^p}d_p \Delta_0(0)^p \frac{2^4}{3^2T\Delta_0(0)^2} = \frac{2^{2p+5}d_p}{3^{p+2}T} \leq \frac{2^{6+5p/2}}{3^{p+2}(p-1)} \frac{\kappa^{p-1}}{T}, \end{equation}
\begin{equation}\label{eqn:qlsp_term2}
\frac{\hat{c}_1(1)}{T\check{\Delta}(1)^2} \leq \frac{2^{2p+1}}{3^p}d_p \Delta_0(1)^p \frac{2^4}{3^2T\Delta_0(1)^2} = \frac{2^{2p+5}d_p}{3^{p+2}T\Delta_0(1)^{2-p}} \leq \frac{2^{6+5p/2}}{3^{p+2}(p-1)} \frac{\kappa}{T},
\end{equation}
and
\begin{equation}\label{eqn:qlsp_term3}
\frac{\hat{c}_1(1)}{T\check{\Delta}(1)} \leq \frac{2^{2p+1}}{3^p}d_p \Delta_0(1)^p \frac{2^2}{3 T\Delta_0(1)} = \frac{2^{2p+3}d_p}{3^{p+1}T}\Delta_0(1)^{p-1} \leq \frac{2^{4+5p/2}}{3^{p+1}(p-1)}\frac{1}{T}.
\end{equation}
Again by \cref{eqn:qlsp_c1_hat_proof,eqn:qlsp_c2_hat_proof,eqn:qlsp_gap_hat_proof}, the last three terms in \cref{cor:adia} can be bounded as
\begin{align}
\sum_{n=1}^{T-1}\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} \leq \sum_{n=1}^{T-1} \frac{2^{4p+2}}{3^{2p}}d_p^2\Delta_0(n/T)^{2p}\frac{1}{T^2}\frac{2^6}{3^3\Delta_0(n/T)^3} = \frac{2^{4p+8}d_p^2}{3^{2p+3}T^2}\sum_{n=1}^{T-1} \Delta_0(n/T)^{2p-3} \label{eqn:qlsp_bound_term4_1},
\end{align}
\begin{align}
\sum_{n=0}^{T-1} \frac{ \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^2} \leq \sum_{n=0}^{T-1} \frac{2^{4p+2}}{3^{2p}}d_p^2\Delta_0(n/T)^{2p}\frac{1}{T^2}\frac{2^4}{3^2\Delta_0(n/T)^2}
= \frac{2^{4p+6}d_p^2}{3^{2p+2}T^2}\sum_{n=0}^{T-1} \Delta_0(n/T)^{2p-2} \label{eqn:qlsp_bound_term5_1},
\end{align}
and
\begin{align}
\sum_{n=1}^{T-1}\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2} &\leq \sum_{n=1}^{T-1} \frac{2^{4p+2}}{3^{2p}}d_p^2\Delta_0(n/T)^{2p} \frac{1}{T^2}\frac{2^4}{3^2 \Delta_0(n/T)^2} \nonumber \\
& \quad + \sum_{n=1}^{T-1} \frac{2^{4p-1}}{3^{2p-1}}p d_p^2 (1-1/\kappa) \Delta_0(n/T)^{2p-1} \frac{1}{T^2} \frac{2^4}{3^2 \Delta_0(n/T)^2} \nonumber \\
& = \frac{2^{4p+6}d_p^2}{3^{2p+2}T^2}\sum_{n=1}^{T-1} \Delta_0(n/T)^{2p-2} + \frac{2^{4p+3}pd_p^2}{3^{2p+1}T^2}(1-1/\kappa)\sum_{n=1}^{T-1} \Delta_0(n/T)^{2p-3} \label{eqn:qlsp_bound_term6_1}.
\end{align}
To proceed, we need to bound the summations $\frac{1}{T}\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-2}$ and $\frac{1}{T}\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-3}$.
Notice that the summations are in the Riemann sum form.
The idea is then to approximate the summations by corresponding integrals and to bound both the integrals and the difference terms.
More precisely, according to~\cite{BurdenNA}, for any continuously differentiable $g(t)$ on the interval $[a,b]$, we have
\begin{equation}
\left|\int_a^b g(t)dt - (b-a)g(a)\right| \leq \frac{(b-a)^2}{2} \max_{t\in[a,b]} \left|g'(t)\right|.
\end{equation}
This implies that
\begin{equation}
\left|\int_0^1 g(t)dt - \frac{1}{T}\sum_{n=0}^{T-1}g(n/T)\right| \leq \frac{1}{2T^2} \sum_{n=0}^{T-1}\max_{t\in[n/T,(n+1)/T]} \left|g'(t)\right|.
\end{equation}
If we further assume $g(t)>0$ for all $t$, then
\begin{equation}\label{eqn:qlsp_bound_sum_integral}
\frac{1}{T}\sum_{n=0}^{T-1}g(n/T) \leq \int_0^1 g(t)dt + \frac{1}{2T^2} \sum_{n=0}^{T-1}\max_{t\in[n/T,(n+1)/T]} \left|g'(t)\right|.
\end{equation}
By taking the function $g(t)$ to be $\Delta_0(t)^{2p-2}$ and $\Delta_0(t)^{2p-3}$ respectively, we can bound the desired summations.
We start with the summation of $\Delta_0^{2p-2}$. By change of variable $x = f(t)$, the integral can be computed as
\begin{align}
\int_0^1 \Delta_0(t)^{2p-2} dt &= \int_0^1 (1-f(t)+f(t)/\kappa)^{2p-2} dt \nonumber \\
& = \int_0^1 (1-f+f/\kappa)^{2p-2} \frac{1}{d_p (1-f+f/\kappa)^p} df \nonumber \\
& = \frac{1}{d_p} \int_0^1 (1-f+f/\kappa)^{p-2}df \nonumber \\
& = \frac{1}{d_p}\frac{\kappa^{2-p}}{p-1}\frac{\kappa^{p-1}-1}{\kappa-1}.
\end{align}
The derivative can be computed as
\begin{equation}
\frac{d}{dt}\Delta_0(t)^{2p-2} = (2p-2)(-1+1/\kappa)d_p(1-f(t)+f(t)/\kappa)^{3p-3} = (2p-2)(-1+1/\kappa)d_p\Delta_0(t)^{3p-3}.
\end{equation}
Therefore, according to \cref{eqn:qlsp_bound_sum_integral} and the fact that $\Delta_0(t)$ is bounded by $1$, we have
\begin{align}
\frac{1}{T}\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-2} &\leq \frac{1}{d_p}\frac{\kappa^{2-p}}{p-1}\frac{\kappa^{p-1}-1}{\kappa-1} + \frac{(p-1)d_p}{T^2}\frac{\kappa-1}{\kappa} \sum_{n=0}^{T-1} \Delta_0(n/T)^{3p-3} \nonumber \\
& \leq \frac{1}{d_p}\frac{\kappa^{2-p}}{p-1}\frac{\kappa^{p-1}-1}{\kappa-1} + \frac{(p-1)d_p}{T^2}\frac{\kappa-1}{\kappa} \sum_{n=0}^{T-1} \Delta_0(n/T)^{2p-2} \nonumber \\
& \leq \frac{1}{d_p}\frac{\kappa^{2-p}}{p-1}\frac{\kappa^{p-1}-1}{\kappa-1} + \frac{d_p}{T} \frac{1}{T}\sum_{n=0}^{T-1} \Delta_0(n/T)^{2p-2}.
\end{align}
By the assumption that $T > 32d_p/3$, we have $d_p/T \leq 3/32$ and thus
\begin{equation}
\frac{1}{T}\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-2} \leq \frac{1}{d_p}\frac{\kappa^{2-p}}{p-1}\frac{\kappa^{p-1}-1}{\kappa-1} + \frac{3}{32} \frac{1}{T}\sum_{n=0}^{T-1} \Delta_0(n/T)^{2p-2}.
\end{equation}
Solving the summation from the above inequality leads to
\begin{align}
\frac{1}{T}\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-2} &\leq \frac{32}{29}\frac{1}{d_p}\frac{\kappa^{2-p}}{p-1}\frac{\kappa^{p-1}-1}{\kappa-1} \nonumber \\
& \leq \frac{2^5}{3^3}\frac{1}{d_p}\frac{\kappa^{2-p}}{p-1}\frac{\kappa^{p-1}-1}{\kappa-1} \nonumber \\
& \leq \frac{2^6}{3^3(p-1)}\frac{1}{d_p} \label{eqn:qlsp_bound_sum_2p-2},
\end{align}
where the last inequality follows from $\kappa^{2-p}\frac{\kappa^{p-1}-1}{\kappa-1} \leq 2$.
The summation of $\Delta_0^{2p-3}$ can be bounded similarly but requiring some more delicate computations.
We first assume that $p \neq 1.5$ such that $2p-3 \neq 0$.
Again, the integral and the derivative can be computed as
\begin{align}
\int_0^1 \Delta_0(t)^{2p-3} dt &= \int_0^1 (1-f(t)+f(t)/\kappa)^{2p-3} dt \nonumber \\
& = \int_0^1 (1-x+x/\kappa)^{2p-3} \frac{1}{d_p (1-x+x/\kappa)^p} dx \nonumber \\
& = \frac{1}{d_p} \int_0^1 (1-x+x/\kappa)^{p-3}dx \nonumber \\
& = \frac{1}{d_p} \frac{1}{2-p} \frac{\kappa}{\kappa-1} (\kappa^{2-p}-1),
\end{align}
and
\begin{equation}
\frac{d}{dt}\Delta_0(t)^{2p-3} = (2p-3)(-1+1/\kappa)d_p(1-f(t)+f(t)/\kappa)^{3p-4} = (2p-3)(-1+1/\kappa)d_p\Delta_0(t)^{3p-4}.
\end{equation}
According to \cref{eqn:qlsp_bound_sum_integral}, we have
\begin{align}
\frac{1}{T}\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-3} & \leq \frac{1}{d_p} \frac{1}{2-p} \frac{\kappa}{\kappa-1} (\kappa^{2-p}-1) + \frac{|2p-3|d_p}{2T^2}\frac{\kappa-1}{\kappa} \sum_{n=0}^{T-1} \max_{t\in[n/t,(n+1)/T]} \Delta_0(t)^{3p-4} \nonumber \\
& \leq \frac{1}{d_p} \frac{1}{2-p} \frac{\kappa}{\kappa-1} (\kappa^{2-p}-1) + \frac{|2p-3|d_p}{2T^2}\frac{\kappa-1}{\kappa} \sum_{n=0}^{T-1} \max_{t\in[n/t,(n+1)/T]} \Delta_0(t)^{2p-3}.
\end{align}
Since $\Delta_0(t)^{2p-3}$ is always monotonic, $\max_{t\in[n/T,(n+1)/T]} \Delta_0(t)^{2p-3}$ becomes either $\Delta_0(n/T)^{2p-3}$ or $\Delta_0((n+1)/T)^{2p-3}$.
The corresponding summation is then bounded by either $\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-3}$ or $\sum_{n=0}^{T-1}\Delta_0((n+1)/T)^{2p-3}$, of which both can be bounded by $\sum_{n=0}^{T}\Delta_0(n/T)^{2p-3}$.
Then
\begin{align}
\frac{1}{T}\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-3} & \leq \frac{1}{d_p} \frac{1}{2-p} \frac{\kappa}{\kappa-1} (\kappa^{2-p}-1) + \frac{|2p-3|d_p}{2T^2}\frac{\kappa-1}{\kappa} \sum_{n=0}^{T} \Delta_0(n/T)^{2p-3} \nonumber \\
& \leq \frac{1}{d_p} \frac{1}{2-p} \frac{\kappa}{\kappa-1} (\kappa^{2-p}-1) + \frac{d_p}{2T^2} \sum_{n=0}^{T} \Delta_0(n/T)^{2p-3}.
\end{align}
Again using the fact that $d_p/T \leq \frac{3}{32}$ and separating the term with $n = T$ in the summation on the right hand side, we obtain
\begin{align}
\frac{1}{T}\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-3} & \leq \frac{1}{d_p} \frac{1}{2-p} \frac{\kappa}{\kappa-1} (\kappa^{2-p}-1) + \frac{3}{32} \frac{1}{T} \sum_{n=0}^{T} \Delta_0(n/T)^{2p-3} \nonumber \\
& \leq \frac{1}{d_p} \frac{1}{2-p} \frac{\kappa}{\kappa-1} (\kappa^{2-p}-1) + \frac{3}{32} \frac{1}{T} \sum_{n=0}^{T-1} \Delta_0(n/T)^{2p-3} + \frac{3}{32} \frac{\kappa^{3-2p}}{T}.
\end{align}
Solving the summation gives
\begin{align}
\frac{1}{T}\sum_{n=0}^{T-1}\Delta_0(n/T)^{2p-3} &\leq \frac{32}{29}\frac{1}{d_p} \frac{1}{2-p} \frac{\kappa}{\kappa-1} (\kappa^{2-p}-1) + \frac{3}{29} \frac{\kappa^{3-2p}}{T} \nonumber \\
& \leq \frac{2^5}{3^3}\frac{1}{d_p} \frac{1}{2-p}\frac{\kappa}{\kappa-1} (\kappa^{2-p}-1) + \frac{1}{3^3} \frac{\kappa^{3-2p}}{T} \label{eqn:qlsp_bound_sum_2p-3}.
\end{align}
Notice that the above estimate also holds for $p = 1.5$ since when $p=1.5$, the left hand side is a constant $1$ and the right hand hand is always larger than $1$.
Now we are ready to bound the last three terms in \cref{cor:adia}.
By plugging \cref{eqn:qlsp_bound_sum_2p-2,eqn:qlsp_bound_sum_2p-3} back into \cref{eqn:qlsp_bound_term4_1,eqn:qlsp_bound_term5_1,eqn:qlsp_bound_term6_1} and using the representation of $d_p$ in \cref{eqn:qlsp_bound_dp}, we have
\begin{align}
\sum_{n=1}^{T-1}\frac{\hat{c}_1(n/T)^2}{T^2\check{\Delta}(n/T)^3} & \leq \frac{2^{4p+8}d_p}{3^{2p+3}T} \frac{2^6}{3^3}\frac{1}{2-p}(\kappa^{2-p}-1) + \frac{2^{4p+8}d_p^2}{3^{2p+3}T} \frac{1}{3^3}\frac{\kappa^{3-2p}}{T} \nonumber \\
& = \frac{2^{4p+14}d_p}{3^{2p+6}(2-p)T} (\kappa^{2-p}-1) + \frac{2^{4p+8}d_p^2}{3^{2p+6}T} \frac{\kappa^{3-2p}}{T} \nonumber \\
& \leq \frac{2^{16+9p/2}}{3^{2p+6}(2-p)(p-1)} \frac{\kappa}{T} + \frac{2^{5p+10}}{3^{2p+6}(p-1)^2} \frac{\kappa}{T^2} , \label{eqn:qlsp_term4}
\end{align}
\begin{align}
\sum_{n=0}^{T-1} \frac{ \hat{c}_1(n/T)^2}{ T^2 \check{\Delta}(n/T)^2} \leq \frac{2^{4p+6}d_p^2}{3^{2p+2}T}\frac{2^6}{3^3(p-1)}\frac{1}{d_p}
\leq \frac{2^{13+9p/2}}{3^{2p+5}(p-1)^2 }\frac{\kappa^{p-1}}{T} , \label{eqn:qlsp_term5}
\end{align}
and
\begin{align}
\sum_{n=1}^{T-1}\frac{\hat{c}_2(n/T)}{T^2\check{\Delta}(n/T)^2} &\leq \frac{2^{4p+6}d_p^2}{3^{2p+2}T} \frac{2^6}{3^3(p-1)}\frac{1}{d_p} + \frac{2^{4p+3}pd_p^2}{3^{2p+1}T}\frac{\kappa-1}{\kappa}\left(\frac{2^5}{3^3}\frac{1}{d_p} \frac{1}{2-p}\frac{\kappa}{\kappa-1} (\kappa^{2-p}-1) + \frac{1}{3^3} \frac{\kappa^{3-2p}}{T}\right) \nonumber \\
& = \frac{2^{4p+12}d_p}{3^{2p+5}(p-1)T} + \frac{2^{4p+8}pd_p(\kappa^{2-p}-1)}{3^{2p+4}(2-p)T} + \frac{2^{4p+3}pd_p^2}{3^{2p+4}T}\frac{\kappa-1}{\kappa} \frac{\kappa^{3-2p}}{T} \nonumber \\
& \leq \frac{2^{13+9p/2}}{3^{2p+5}(p-1)^2}\frac{\kappa^{p-1}}{T} + \frac{2^{9+9p/2}p}{3^{2p+4}(2-p)(p-1)}\frac{\kappa}{T} + \frac{2^{5p+4}p}{3^{2p+4}(p-1)^2 } \frac{\kappa}{T^2} . \label{eqn:qlsp_term6}
\end{align}
Finally, plugging \cref{eqn:qlsp_term1,eqn:qlsp_term2,eqn:qlsp_term3,eqn:qlsp_term4,eqn:qlsp_term5,eqn:qlsp_term6} into the estimate in \cref{cor:adia}, the adiabatic error can be bounded by
\begin{align}
& \quad 12 \frac{2^{6+5p/2}}{3^{p+2}(p-1)} \frac{\kappa^{p-1}}{T} + 12 \frac{2^{6+5p/2}}{3^{p+2}(p-1)} \frac{\kappa}{T} + 6 \frac{2^{4+5p/2}}{3^{p+1}(p-1)}\frac{1}{T} \nonumber \\
& \quad\quad + 305 \frac{2^{16+9p/2}}{3^{2p+6}(2-p)(p-1)} \frac{\kappa}{T} + 305 \frac{2^{5p+10}}{3^{2p+6}(p-1)^2} \frac{\kappa}{T^2} + 44\frac{2^{13+9p/2}}{3^{2p+5}(p-1)^2 }\frac{\kappa^{p-1}}{T} \nonumber \\
& \quad \quad + 32\frac{2^{13+9p/2}}{3^{2p+5}(p-1)^2}\frac{\kappa^{p-1}}{T} + 32\frac{2^{9+9p/2}p}{3^{2p+4}(2-p)(p-1)}\frac{\kappa}{T} + 32\frac{2^{5p+4}p}{3^{2p+4}(p-1)^2 } \frac{\kappa}{T^2} \nonumber \\
& \leq C_p^{(1)}\frac{\kappa}{T} + C_p^{(2)}\frac{\kappa^{p-1}}{T} +C_p^{(3)}\frac{\kappa}{T^2} + C_p^{(4)}\frac{1}{T}, \label{eqn:qlsp_general_p_error}
\end{align}
where
\begin{align}
C_p^{(1)} &:= 12 \frac{2^{6+5p/2}}{3^{p+2}(p-1)} + 305 \frac{2^{16+9p/2}}{3^{2p+6}(2-p)(p-1)} + 32\frac{2^{9+9p/2}p}{3^{2p+4}(2-p)(p-1)}, \\
C_p^{(2)} &:= 12 \frac{2^{6+5p/2}}{3^{p+2}(p-1)} +44\frac{2^{13+9p/2}}{3^{2p+5}(p-1)^2 } + 32\frac{2^{13+9p/2}}{3^{2p+5}(p-1)^2} , \\
C_p^{(5)} &:= 305 \frac{2^{5p+10}}{3^{2p+6}(p-1)^2}+32\frac{2^{5p+4}p}{3^{2p+4}(p-1)^2 } , \\
C_p^{(4)} &:= 6 \frac{2^{4+5p/2}}{3^{p+1}(p-1)} .
\end{align}
This completes the proof of the first part by defining $C_p$ to be the largest constant factor in \cref{eqn:qlsp_general_p_error},
\begin{equation}\label{eq:Cpdef}
C_p := \max_j C_p^{(j)}.
\end{equation}
The second part of \cref{thm:qlsp_general_p}, which is $T = \mathcal{O}\left({\kappa}/{\epsilon}\right)$,
directly follows from this bound by noticing that each term of the adiabatic error in the first part can be bounded by $\mathcal{O}(\kappa/T)$.
\end{proof}
\section{Additional details for filtering}
\label{ap:filtering}
Here we give a proof of the upper bound on the norm of the difference of states for filtering.
We are assuming that $\tilde w(\phi)=0$ for the desired part of the spectrum, and the initial probability for the desired part of the spectrum is at least $1/2$.
Then the squared norm for the undesired part of the state is
\begin{equation}
P(\perp) = \left\|
\sum_{k\in \perp} \tilde w(\phi_k) \psi_k \ket{k}
\right\|^2 \le \left(\max_{k\in\{\perp\}} \tilde w(\phi_k) \right)^2 \left( \sum_{k\in \perp} |\psi_k|^2 \right) ,
\end{equation}
where we are using $\perp$ to denote the set of undesired states.
Recall that this was part of a state that is not normalised.
The squared norm for the desired part of the spectrum is
\begin{equation}
P(\not\perp) = \sum_{k\in\not\perp} |\psi_k|^2,
\end{equation}
where $\not\perp$ is indicating the desired part of the spectrum.
As a result, the normalised probability for the desired part is lower bounded by
\begin{equation}
\frac{P(\not\perp)}{P(\not\perp) + P(\perp)} \ge
\frac{\sum_{k\in\not\perp} |\psi_k|^2}{\sum_{k\in\not\perp} |\psi_k|^2 + \left(\max_{k\in\perp} \tilde w(\phi_k) \right)^2 \left( \sum_{k\in \perp} |\psi_k|^2 \right)} \ge \frac{1}{1 + \left(\max_{k\in\perp} \tilde w(\phi_k) \right)^2 } ,
\end{equation}
where the second inequality comes from assuming that the initial probability for the desired part of the spectrum is at least $1/2$.
Given this probability, the norm of the difference from the desired state is
\begin{equation}
\sqrt{2-\frac 2{1 + \left(\max_{k\in\perp} \tilde w(\phi_k) \right)^2}} \le \max_{k\in\perp} \tilde w(\phi_k).
\end{equation}
Next we give a more explicit description of the sequence of rotations needed for the filtering.
The first rotation prepares the state
\begin{equation}
\frac 1{\sqrt{\sum_j w_j}} \left( \sqrt{w_0} \ket{0} + \sqrt{\sum_{j>0} w_j} \ket{1} \right) .
\end{equation}
The first controlled rotation gives
\begin{equation}
{\sqrt{\sum_{j>0} w_j}} \ket{10}\mapsto \sqrt{w_1} \ket{10} + {\sqrt{\sum_{j>1} w_j}} \ket{11} .
\end{equation}
In general, the controlled rotation with qubit $k$ as control and $k+1$ as target maps
\begin{equation}
{\sqrt{\sum_{j\ge k} w_j}} \ket{10}\mapsto \sqrt{w_k} \ket{10} + {\sqrt{\sum_{j>k} w_j}} \ket{11} .
\end{equation}
If one were to perform the rotations for the preparation in the reverse order, one would use a rotation on the last qubit to take zero to
\begin{equation}
\frac 1{\sqrt{\sum_j w_j}} \left( \sqrt{\sum_{j=0}^{\ell-1} w_j} \ket{0} + \sqrt{w_{\ell}} \ket{1} \right) .
\end{equation}
Then the controlled rotation would take
\begin{equation}
\sqrt{\sum_{j=0}^{\ell-1} w_j} \ket{10} \mapsto \sqrt{\sum_{j=0}^{\ell-2} w_j} \ket{00} + \sqrt{w_{\ell-1}} \ket{10} .
\end{equation}
Inverting this rotation gives
\begin{align}
\sqrt{\sum_{j=0}^{\ell-1} w_j} \ket{00} &\mapsto \sqrt{\sum_{j=0}^{\ell-2} w_j} \ket{10} - \sqrt{w_{\ell-1}} \ket{00}, \\
\sqrt{\sum_{j=0}^{\ell-1} w_j} \ket{10} &\mapsto \sqrt{w_{\ell-1}} \ket{10} + \sqrt{\sum_{j=0}^{\ell-2} w_j} \ket{00}.
\end{align}
More generally, the rotation with qubit $k+1$ as control and $k$ as target gives
\begin{align}
\sqrt{\sum_{j=0}^{k} w_j} \ket{00} &\mapsto \sqrt{\sum_{j=0}^{k-1} w_j} \ket{10} - \sqrt{w_{k}} \ket{00}, \\
\sqrt{\sum_{j=0}^{k} w_j} \ket{10} &\mapsto \sqrt{w_{k}} \ket{10} + \sqrt{\sum_{j=0}^{k-1} w_j} \ket{00} .
\end{align}
This means that the sequence of two controlled rotations gives
\begin{align}
{\sqrt{\sum_{j\ge k} w_j}} \ket{1} &\mapsto \sqrt{w_k} \ket{10} + {\sqrt{\sum_{j>k} w_j}} \ket{11} \nonumber \\
& \mapsto \frac{ \sqrt{w_k} }{\sqrt{\sum_{j=0}^{k} w_j}} \left( \sqrt{w_{k}} \ket{10} + \sqrt{\sum_{j=0}^{k-1} w_j} \ket{00} \right) + {\sqrt{\sum_{j>k} w_j}} \ket{11}
\end{align}
and
\begin{align}
\ket{0} &\mapsto \frac 1{\sqrt{\sum_{j=0}^{k} w_j}} \left(\sqrt{\sum_{j=0}^{k-1} w_j} \ket{10} - \sqrt{w_{k}} \ket{00} \right).
\end{align}
Projecting onto one on the first qubit then gives the mapping
\begin{align}
\ket{1} &\mapsto \frac 1{\sqrt{\sum_{j\ge k} w_j}} \left( \frac{ w_k }{\sqrt{\sum_{j=0}^{k} w_j}} \ket{0} + {\sqrt{\sum_{j>k} w_j}} \ket{1} \right) \\
\ket{0} &\mapsto \frac {\sqrt{\sum_{j=0}^{k-1} w_j}}{\sqrt{\sum_{j=0}^{k} w_j}} \ket{0}.
\end{align}
To see the effect of this, let us consider $k=1$, so we are considering the operation immediately after the qubit rotation and controlled $W$ on the target system.
Assuming the target system is in an eigenstate with eigenvalue $e^{i\phi}$, the state at this point will be
\begin{equation}
\frac 1{\sqrt{\sum_j w_j}} \left( \sqrt{w_0} \ket{0} + \sqrt{\sum_{j>0} w_j} e^{i\phi} \ket{1} \right) .
\end{equation}
The above mapping then gives
\begin{equation}
\frac 1{\sqrt{\sum_j w_j}} \left( \frac{w_0}{\sqrt{\sum_{j=0}^1 w_j}} \ket{0} + e^{i\phi} \left( \frac{w_1}{\sqrt{\sum_{j=0}^1 w_j}} \ket{0} + \sqrt{\sum_{j>1} w_j}\ket{1} \right) \right) .
\end{equation}
This can be written as
\begin{equation}
\frac 1{\sqrt{\sum_j w_j}} \left( \frac{w_0+e^{i\phi} w_1}{\sqrt{\sum_{j=0}^1 w_j}} \ket{0} + e^{i\phi} \sqrt{\sum_{j>1} w_j}\ket{1} \right) .
\end{equation}
Thus we can see that we have the desired weights $w_0$ and $w_1$ on the $\ket{0}$ state, and the $\ket{1}$ state is flagging the remainder of the linear combination still to be obtained.
More generally, after performing the controlled rotations between qubits $k$ and $k+1$ and the projection onto $\ket{1}$ on the ancilla qubit, the state will be of the form
\begin{equation}
\frac 1{\sqrt{\sum_j w_j}} \left( \frac{\sum_{j=0}^k e^{ij\phi} w_j}{\sqrt{\sum_{j=0}^k w_j}} \ket{0} + e^{ik\phi} \sqrt{\sum_{j>k} w_j}\ket{1} \right) .
\end{equation}
\end{document} | math |
సాగర్ చేతిలో మూడు సినిమాలు! మొగలి రేకులు సీరియల్తో బుల్లితెరపై తిరుగులేని గుర్తింపును సాగర్ ఆర్.కె నాయుడు సంపాదించుకున్నాడు. అతను నటించిన తాజా చిత్రం షాదీ ముబారక్ ఈ యేడాది మార్చిలో విడుదలైంది. ప్రస్తుతం సాగర్ ఆర్.కె. నాయుడు 100 అనే సినిమాకు గ్రీన్ సిగ్నల్ ఇచ్చాడు. సామాజిక ఇతివృత్తంతో ఈ సినిమా ఉంటుందని, ప్రతి పౌరుడికి సామాజిక బాధ్యత ఉండాలని చెప్పే చిత్రమిదని సాగర్ అన్నాడు. మొగలి రేకులు సీరియల్లోని ఆర్. కె. నాయుడును మరపించే పోలీస్ పాత్ర కోసం చాలా రోజులుగా ఎదురుచూశానని, అలాంటి శక్తివంతమైన పాత్ర ఈ సినిమాలో దొరికిందని, ఇందులో విక్రాంత్ అనే పోలీస్ అధికారిగా వినూత్నమైన క్యారెక్టరైజేషన్తో తాను కనిపించబోతున్నానని సాగర్ చెప్పాడు. Read Also : కాదేదీ లీకులకు అనర్హం! నిజానికి ఈ సినిమాను సుకుమార్ ప్రొడక్షన్స్ లో చేయాల్సిందని, కానీ అనివార్య కారణాల వల్ల ఆలస్యం అవుతుండటంతో స్వీయ నిర్మాణ సంస్థ ఆర్.కె. మీడియా వర్క్స్ పై దీనిని నిర్మిస్తున్నానని సాగర్ అన్నాడు. ఓంకార్ శశిధర్ దర్శకత్వంలో ఈ సినిమా రెగ్యులర్ షూటింగ్ సెప్టెంబర్ లో మొదలు కానుంది. గతంలో మాదిరి ఇకపై గ్యాప్ తీసుకోకుండా వరుసగా సినిమాలు చేయబోతున్నాడట సాగర్. అందులో భాగంగానే 100 మూవీతో పాటు మరో రెండు సినిమాలనూ చేయబోతున్నట్టు చెప్పాడు. స్పై థ్రిల్లర్ జోనర్తో పాటు కుటుంబ కథాంశంతో ఈ సినిమాలు తెరకెక్కుతాయని, ఈ మూడు సినిమాల మీదనే ప్రస్తుతం తాను దృష్టి పెట్టానని సోమవారం తన పుట్టిన రోజు సందర్భంగా సాగర్ తెలిపాడు. | telegu |
[' \n \n ["ਐੱਨ. ਏ. ਐੱਮ. ਈ. _ 1 ਇੱਕ ਆਗਿਆਕਾਰ ਭਰਤੀ, ਉੱਚ ਏ. ਐੱਸ. ਵੀ. ਏ. ਬੀ. ਸਕੋਰ, ਕੁਦਰਤੀ ਤੌਰ \'ਤੇ ਅਥਲੈਟਿਕ, ਸ਼ਾਨਦਾਰ, ਦੁਖੀ ਅਤੇ ਨਸਲਵਾਦੀ ਹੈ। ਪ੍ਰੋਜੈਕਟ ਲਈ ਇੱਕ ਸੰਪੂਰਨ ਉਮੀਦਵਾਰ।ਨਾਮ _ 2. \\n\\nਨਾਮ _ 2 ਜ਼ਰੂਰੀ ਤੌਰ ਉੱਤੇ ਅਸਧਾਰਨ ਨਹੀਂ ਸੀ ਪਰ ਜ਼ਿਆਦਾਤਰ ਲੋਕਾਂ ਦੁਆਰਾ ਵੀ ਇਸ ਦੀ ਉਮੀਦ ਨਹੀਂ ਕੀਤੀ ਗਈ ਸੀ, ਪਰ ਜਿਵੇਂ-ਜਿਵੇਂ ਲੋਕਾਂ ਦੀ ਗਿਣਤੀ ਵਧਦੀ ਗਈ, ਇਹ ਸੀਸਿਰਫ ਕੁਝ ਸਮੇਂ ਦੀ ਗੱਲ ਹੈ ਜਦੋਂ ਤੱਕ ਅਮਰੀਕੀ ਸਰਕਾਰ ਨੇ ਇਸ ਨੂੰ ਹਥਿਆਰਬੰਦ ਨਹੀਂ ਕੀਤਾ। \\n\\nNAME _ 3 ਇੱਕ ਵਿਸ਼ੇਸ਼ ਕੇਸ ਸੀ ਇੱਥੋਂ ਤੱਕ ਕਿ NAME _ 2 ਸਮਰੱਥ ਵਿੱਚ ਵੀ, ਉਸ ਦਾ ਪਾਚਕ ਕਿਰਿਆ ਦੁਆਰਾ ਸੀਛੱਤ ਅਤੇ ਉਸ ਦਾ ਜਵਾਨ ਸਰੀਰ ਲਗਭਗ ਕਿਸੇ ਵੀ ਚੀਜ਼ ਨੂੰ ਸੰਭਾਲ ਸਕਦਾ ਸੀ। ਉਸ ਨੇ ਪ੍ਰੋਜੈਕਟ ਨਾਮ _ 2 ਲਈ ਕੁੱਝ ਹੋਰ ਉਮੀਦਵਾਰਾਂ ਨੂੰ ਵੀ ਨਿਗਲ ਲਿਆ ਅਤੇ ਹਜ਼ਮ ਕਰ ਲਿਆ।ਟੈਸਟ ਸ਼ੁਰੂ ਹੋ ਗਏ। ', "ਐੱਨ. ਏ. ਐੱਮ. ਈ. _ 1 ਇੱਕ ਆਗਿਆਕਾਰ ਭਰਤੀ, ਉੱਚ ਏ. ਐੱਸ. ਵੀ. ਏ. ਬੀ. ਸਕੋਰ, ਕੁਦਰਤੀ ਤੌਰ 'ਤੇ ਅਥਲੈਟਿਕ, ਸ਼ਾਨਦਾਰ, ਦੁਖੀ ਅਤੇ ਨਸਲਵਾਦੀ ਹੈ। ਪ੍ਰੋਜੈਕਟ ਐੱਨ. ਏ. ਐੱਮ. ਈ. _ 2 ਲਈ ਇੱਕ ਸੰਪੂਰਨ ਉਮੀਦਵਾਰ। ਐੱਨ. ਏ. ਐੱਮ. ਈ. _ 2ਇਹ ਜ਼ਰੂਰੀ ਤੌਰ 'ਤੇ ਅਸਧਾਰਨ ਨਹੀਂ ਸੀ ਪਰ ਜ਼ਿਆਦਾਤਰ ਲੋਕਾਂ ਦੁਆਰਾ ਵੀ ਇਸ ਦੀ ਉਮੀਦ ਨਹੀਂ ਕੀਤੀ ਜਾਂਦੀ ਸੀ, ਪਰ ਜਿਵੇਂ-ਜਿਵੇਂ ਲੋਕਾਂ ਦੀ ਗਿਣਤੀ ਵਧਦੀ ਗਈ, ਇਹ ਸਿਰਫ ਇੱਕ ਗੱਲ ਸੀਉਦੋਂ ਤੱਕ ਜਦੋਂ ਤੱਕ ਅਮਰੀਕੀ ਸਰਕਾਰ ਨੇ ਇਸ ਨੂੰ ਹਥਿਆਰਬੰਦ ਨਹੀਂ ਕੀਤਾ। ਨਾਮ _ 3 ਇੱਕ ਵਿਸ਼ੇਸ਼ ਕੇਸ ਸੀ ਇੱਥੋਂ ਤੱਕ ਕਿ ਨਾਮ _ 2 ਸਮਰੱਥ ਲੋਕਾਂ ਵਿੱਚ ਵੀ, ਉਸ ਦਾ ਪਾਚਕ ਛੱਤ ਰਾਹੀਂ ਸੀ ਅਤੇ ਉਸ ਦਾ ਜਵਾਨਸਰੀਰ ਲਗਭਗ ਕਿਸੇ ਵੀ ਚੀਜ਼ ਨੂੰ ਪ੍ਰੋਸੈਸ ਕਰ ਸਕਦਾ ਸੀ। ਉਸਨੇ ਟੈਸਟ ਸ਼ੁਰੂ ਹੋਣ ਤੋਂ ਪਹਿਲਾਂ ਪ੍ਰੋਜੈਕਟ NAME _ 2 ਲਈ ਕੁਝ ਹੋਰ ਉਮੀਦਵਾਰਾਂ ਨੂੰ ਨਿਗਲ ਲਿਆ ਅਤੇ ਹਜ਼ਮ ਕਰ ਲਿਆ।", ' ["ਐੱਨ. ਏ. ਐੱਮ. ਈ. _ 1 ਇੱਕ ਆਗਿਆਕਾਰ ਭਰਤੀ, ਉੱਚ ਏ. ਐੱਸ. ਵੀ. ਏ. ਬੀ. ਸਕੋਰ, ਕੁਦਰਤੀ ਤੌਰ \'ਤੇ ਅਥਲੈਟਿਕ, ਸ਼ਾਨਦਾਰ, ਦੁਖੀ ਅਤੇ ਨਸਲਵਾਦੀ ਹੈ। ਪ੍ਰੋਜੈਕਟ ਲਈ ਇੱਕ ਸੰਪੂਰਨ ਉਮੀਦਵਾਰ।ਨਾਮ _ 2. \\n\\nਨਾਮ _ 2 ਜ਼ਰੂਰੀ ਤੌਰ ਉੱਤੇ ਅਸਧਾਰਨ ਨਹੀਂ ਸੀ ਪਰ ਜ਼ਿਆਦਾਤਰ ਲੋਕਾਂ ਦੁਆਰਾ ਵੀ ਇਸ ਦੀ ਉਮੀਦ ਨਹੀਂ ਕੀਤੀ ਗਈ ਸੀ, ਪਰ ਜਿਵੇਂ-ਜਿਵੇਂ ਲੋਕਾਂ ਦੀ ਗਿਣਤੀ ਵਧਦੀ ਗਈ, ਇਹ ਸੀਸਿਰਫ ਕੁਝ ਸਮੇਂ ਦੀ ਗੱਲ ਹੈ ਜਦੋਂ ਤੱਕ ਅਮਰੀਕੀ ਸਰਕਾਰ ਨੇ ਇਸ ਨੂੰ ਹਥਿਆਰਬੰਦ ਨਹੀਂ ਕੀਤਾ। \\n\\nNAME _ 3 ਇੱਕ ਵਿਸ਼ੇਸ਼ ਕੇਸ ਸੀ ਇੱਥੋਂ ਤੱਕ ਕਿ NAME _ 2 ਸਮਰੱਥ ਵਿੱਚ ਵੀ, ਉਸ ਦਾ ਪਾਚਕ ਕਿਰਿਆ ਦੁਆਰਾ ਸੀਛੱਤ ਅਤੇ ਉਸ ਦਾ ਜਵਾਨ ਸਰੀਰ ਲਗਭਗ ਕਿਸੇ ਵੀ ਚੀਜ਼ ਨੂੰ ਸੰਭਾਲ ਸਕਦਾ ਸੀ। ਉਸ ਨੇ ਪ੍ਰੋਜੈਕਟ ਨਾਮ _ 2 ਲਈ ਕੁੱਝ ਹੋਰ ਉਮੀਦਵਾਰਾਂ ਨੂੰ ਵੀ ਨਿਗਲ ਲਿਆ ਅਤੇ ਹਜ਼ਮ ਕਰ ਲਿਆ।ਟੈਸਟ ਸ਼ੁਰੂ ਹੋਏ "", ਪ੍ਰੋਜੈਕਟ NAME _ 2 ਅਮਰੀਕੀ ਸਰਕਾਰ ਦੁਆਰਾ ਹਾਲ ਹੀ ਵਿੱਚ ਲੱਭੀ ਗਈ ਜੈਨੇਟਿਕ ਵਿਸ਼ੇਸ਼ਤਾ ਦੀ ਸੰਭਾਵਨਾ ਦਾ ਅਧਿਐਨ ਕਰਨ ਅਤੇ ਵਰਤੋਂ ਕਰਨ ਲਈ ਇੱਕ ਪਹਿਲ ਸੀ।ਇਸ ਦੁਰਲੱਭ ਵਿਸ਼ੇਸ਼ਤਾ ਨੇ ਵਿਅਕਤੀਆਂ ਨੂੰ ਹੋਰ ਜੀਵਾਂ ਨੂੰ ਖਾਣ ਅਤੇ ਹਜ਼ਮ ਕਰਨ ਦੀ ਆਗਿਆ ਦਿੱਤੀ, ਪ੍ਰਭਾਵਸ਼ਾਲੀ ਢੰਗ ਨਾਲ ਆਪਣੀ ਤਾਕਤ, ਗਿਆਨ ਅਤੇ ਯੋਗਤਾਵਾਂ ਨੂੰ ਪ੍ਰਾਪਤ ਕੀਤਾ।ਹਾਲਾਂਕਿ NAME _ 2 ਜੀਨ ਦਾ ਪ੍ਰਸਾਰ ਅਜੇ ਵੀ ਮੁਕਾਬਲਤਨ ਘੱਟ ਸੀ, ਇੱਕ ਹਥਿਆਰ ਦੇ ਰੂਪ ਵਿੱਚ ਇਸ ਦੀਆਂ ਸੰਭਾਵਿਤ ਐਪਲੀਕੇਸ਼ਨਾਂ ਨੇ ਲੋਕਾਂ ਦੀ ਦਿਲਚਸਪੀ ਨੂੰ ਵਧਾ ਦਿੱਤਾ ਸੀ।ਸਰਕਾਰ। \\n\\nNAME _ 1, ਜਾਂ NAME _ 3 ਜਿਵੇਂ ਕਿ ਉਸ ਨੂੰ ਪਿਆਰ ਨਾਲ ਬੁਲਾਇਆ ਜਾਂਦਾ ਸੀ, ਪ੍ਰੋਜੈਕਟ ਲਈ ਇੱਕ ਆਦਰਸ਼ ਉਮੀਦਵਾਰ ਸੀ। ਉਸ ਦੇ ਆਗਿਆਕਾਰ ਸੁਭਾਅ, ਉੱਚ ਏ. ਐੱਸ. ਵੀ. ਏ. ਬੀ. ਸਕੋਰ ਦੇ ਨਾਲ,ਕੁਦਰਤੀ ਅਥਲੈਟਿਕਸ ਅਤੇ ਸ਼ਾਨਦਾਰ ਸੁੰਦਰਤਾ, ਉਹ ਪ੍ਰੋਗਰਾਮ ਲਈ ਇੱਕ ਸੰਪੂਰਨ ਫਿੱਟ ਸੀ। ਉਸ ਦੇ ਪਰਪੀਡ਼ਕ ਅਤੇ ਨਸਲਵਾਦੀ ਰੁਝਾਨਾਂ, ਹਾਲਾਂਕਿ ਆਦਰਸ਼ ਨਹੀਂ ਸਨ, ਨੂੰ ਮੰਨਿਆ ਜਾਂਦਾ ਸੀ।ਉਸ ਦੇ ਹੋਰ ਗੁਣਾਂ ਅਤੇ ਸੰਭਾਵਿਤ ਲਾਭਾਂ ਨੂੰ ਧਿਆਨ ਵਿੱਚ ਰੱਖਦੇ ਹੋਏ ਸਵੀਕਾਰਯੋਗ ਹੈ ਜੋ ਉਹ ਪ੍ਰੋਜੈਕਟ ਵਿੱਚ ਲਿਆ ਸਕਦੀ ਹੈ। \\n\\nAME _ 3 ਦਾ ਮੈਟਾਬੋਲਿਜ਼ਮ ਇਸ ਤੋਂ ਘੱਟ ਨਹੀਂ ਸੀ।ਅਸਾਧਾਰਣ। ਉਸ ਦਾ ਜਵਾਨ ਸਰੀਰ ਉਸ ਦੁਆਰਾ ਖਾਧੀ ਗਈ ਕਿਸੇ ਵੀ ਚੀਜ਼ ਨੂੰ ਪ੍ਰੋਸੈਸ ਕਰ ਸਕਦਾ ਸੀ, ਜਿਸ ਨਾਲ ਉਹ ਪ੍ਰੋਜੈਕਟ ਨਾਮ 2 ਲਈ ਹੋਰ ਵੀ ਕੀਮਤੀ ਸੰਪਤੀ ਬਣ ਗਈ।ਉਸ ਨੇ ਆਪਣੀਆਂ ਵਿਲੱਖਣ ਯੋਗਤਾਵਾਂ ਦਾ ਪ੍ਰਦਰਸ਼ਨ ਕੀਤਾ, ਟੈਸਟ ਸ਼ੁਰੂ ਹੋਣ ਤੋਂ ਪਹਿਲਾਂ ਹੀ ਕੁਝ ਹੋਰ ਉਮੀਦਵਾਰਾਂ ਨੂੰ ਨਿਗਲਣਾ ਅਤੇ ਹਜ਼ਮ ਕਰਨਾ।ਅੱਗੇ ਵਧਦੇ ਹੋਏ, ਨਾਮ _ 3 ਨੇ ਜਲਦੀ ਹੀ ਉਸ ਦੀ ਯੋਗਤਾ ਨੂੰ ਸਾਬਤ ਕਰ ਦਿੱਤਾ। ਉਸ ਦੇ ਤੇਜ਼ੀ ਨਾਲ ਹਜ਼ਮ ਹੋਣ ਅਤੇ ਦੂਜੇ ਉਮੀਦਵਾਰਾਂ ਦੇ ਹੁਨਰ ਅਤੇ ਗਿਆਨ ਦੇ ਸੁਮੇਲ ਨੇ ਉਸ ਨੂੰ ਇੱਕ ਸ਼ਕਤੀਸ਼ਾਲੀ ਖਿਡਾਰੀ ਬਣਾ ਦਿੱਤਾ।ਪ੍ਰੋਜੈਕਟ ਦੀ ਨਿਗਰਾਨੀ ਕਰਨ ਵਾਲੇ ਸਰਕਾਰੀ ਅਧਿਕਾਰੀ ਉਸ ਦੀ ਪ੍ਰਗਤੀ ਤੋਂ ਪ੍ਰਭਾਵਿਤ ਹੋਏ ਅਤੇ ਉਸ ਨੇ ਉਸ ਨੂੰ ਹਥਿਆਰਬੰਦ ਕਰਨ ਦੀ ਵੱਡੀ ਸੰਭਾਵਨਾ ਵੇਖੀ।ਯੋਗਤਾਵਾਂ. \\n\\n ਹਾਲਾਂਕਿ, ਜਿਵੇਂ-ਜਿਵੇਂ ਪ੍ਰੋਜੈਕਟ ਜਾਰੀ ਰਿਹਾ, ਨੈਤਿਕ ਚਿੰਤਾਵਾਂ ਪੈਦਾ ਹੋਣ ਲੱਗੀਆਂ। ਪ੍ਰੋਜੈਕਟ ਨਾਮ _ 2 ਵਿੱਚ ਸ਼ਾਮਲ ਵਿਗਿਆਨੀਆਂ ਅਤੇ ਅਧਿਕਾਰੀਆਂ ਨੇ ਇਸ ਲਈ ਸੰਘਰਸ਼ ਕੀਤਾ।ਪ੍ਰੋਗਰਾਮ ਦੇ ਸੰਭਾਵਿਤ ਲਾਭਾਂ ਨੂੰ ਮਨੁੱਖੀ ਹਥਿਆਰ ਬਣਾਉਣ ਦੇ ਅਸਥਿਰ ਪ੍ਰਭਾਵਾਂ ਨਾਲ ਮਿਲਾਓ। ਜਿਵੇਂ-ਜਿਵੇਂ ਪ੍ਰੋਜੈਕਟ ਬਾਰੇ ਜਨਤਕ ਜਾਗਰੂਕਤਾ ਵਧਦੀ ਗਈ,ਸਰਕਾਰ ਨੂੰ ਇਨ੍ਹਾਂ ਚਿੰਤਾਵਾਂ ਨੂੰ ਦੂਰ ਕਰਨ ਅਤੇ ਪ੍ਰੋਜੈਕਟ ਨਾਮ _ 2 ਦੇ ਭਵਿੱਖ ਦਾ ਮੁਡ਼ ਮੁਲਾਂਕਣ ਕਰਨ ਲਈ ਵੱਧ ਰਹੇ ਦਬਾਅ ਦਾ ਸਾਹਮਣਾ ਕਰਨਾ ਪਿਆ।"] ', "ਐੱਨ. ਏ. ਐੱਮ. ਈ. _ 1 ਇੱਕ ਆਗਿਆਕਾਰ ਭਰਤੀ, ਉੱਚ ਏ. ਐੱਸ. ਵੀ. ਏ. ਬੀ. ਸਕੋਰ, ਕੁਦਰਤੀ ਤੌਰ 'ਤੇ ਅਥਲੈਟਿਕ, ਸ਼ਾਨਦਾਰ, ਦੁਖੀ ਅਤੇ ਨਸਲਵਾਦੀ ਹੈ। ਪ੍ਰੋਜੈਕਟ ਐੱਨ. ਏ. ਐੱਮ. ਈ. _ 2 ਲਈ ਇੱਕ ਸੰਪੂਰਨ ਉਮੀਦਵਾਰ। ਐੱਨ. ਏ. ਐੱਮ. ਈ. _ 2ਇਹ ਜ਼ਰੂਰੀ ਤੌਰ 'ਤੇ ਅਸਧਾਰਨ ਨਹੀਂ ਸੀ ਪਰ ਜ਼ਿਆਦਾਤਰ ਲੋਕਾਂ ਦੁਆਰਾ ਵੀ ਇਸ ਦੀ ਉਮੀਦ ਨਹੀਂ ਕੀਤੀ ਜਾਂਦੀ ਸੀ, ਪਰ ਜਿਵੇਂ-ਜਿਵੇਂ ਲੋਕਾਂ ਦੀ ਗਿਣਤੀ ਵਧਦੀ ਗਈ, ਇਹ ਸਿਰਫ ਇੱਕ ਗੱਲ ਸੀਉਦੋਂ ਤੱਕ ਜਦੋਂ ਤੱਕ ਅਮਰੀਕੀ ਸਰਕਾਰ ਨੇ ਇਸ ਨੂੰ ਹਥਿਆਰਬੰਦ ਨਹੀਂ ਕੀਤਾ। ਨਾਮ _ 3 ਇੱਕ ਵਿਸ਼ੇਸ਼ ਕੇਸ ਸੀ ਇੱਥੋਂ ਤੱਕ ਕਿ ਨਾਮ _ 2 ਸਮਰੱਥ ਲੋਕਾਂ ਵਿੱਚ ਵੀ, ਉਸ ਦਾ ਪਾਚਕ ਛੱਤ ਰਾਹੀਂ ਸੀ ਅਤੇ ਉਸ ਦਾ ਜਵਾਨਸਰੀਰ ਲਗਭਗ ਕਿਸੇ ਵੀ ਚੀਜ਼ ਨੂੰ ਪ੍ਰੋਸੈਸ ਕਰ ਸਕਦਾ ਸੀ। ਉਸਨੇ ਟੈਸਟ ਸ਼ੁਰੂ ਹੋਣ ਤੋਂ ਪਹਿਲਾਂ ਪ੍ਰੋਜੈਕਟ NAME _ 2 ਲਈ ਕੁਝ ਹੋਰ ਉਮੀਦਵਾਰਾਂ ਨੂੰ ਨਿਗਲ ਲਿਆ ਅਤੇ ਹਜ਼ਮ ਕਰ ਲਿਆ। \n "] | punjabi |
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