text:
\section{Introduction}
\label{sec:introduction}

There has been much recent work on approaches to experimentally engineer many-body quantum phases of matter~\cite{Ioffe2002,Buchler2005,Greiner2002a,Jaksch2005,Micheli2006a,Pupillo2008a,Weimer2010a}. In particular there is a wide array of lattice Hamiltonians whose ground states are novel quantum phases that are not yet known to exist in natural systems~\cite{Wen2003,Moessner2001d,Levin2005a,Kitaev2003,Kitaev2006a}; to physically realize such phases, one must generate the models artificially. Additionally, many of these novel phases have potential applications to quantum information~\cite{Dennis2002,Freedman2002a,Kitaev2003,Nayak2008}. A robust experimental realization of such phases of matter might be a route to building a fault-tolerant quantum computer~\cite{Preskill1998}.

One such class of lattice Hamiltonians are quantum stabilizer Hamiltonians~\cite{Bacon2006,Bombin2011,Bombin2008,Bombin2007,Bombin2006,Bravyi1998,Dennis2002,Kitaev2003,Yoshida2011a,Raussendorf2001}. Stabilizer Hamiltonians are composed of commuting multi-qubit Pauli operators and play a key role in quantum error correction.  Encoding quantum information into ground states of stabilizer Hamiltonians provides a natural physical architecture for realization of quantum error correcting codes and quantum memory in qubit arrays~\cite{Dennis2002}. Because of their characteristic degenerate ground states and finite energy gaps to local excitations, they offer physical protection for encoded quantum information. Much recent interest  has therefore focused on the quantum phases generated by stabilizer Hamiltonians, their ability to generate topological order and their relation to novel exotic phases~\cite{Aguado2008a,Weimer2010a,Verstraete2009a,Brennen2009b}.

While the quantum codes derived from ground states of stabilizer Hamiltonians are of broad theoretical interest in quantum information theory, the transition from theoretical characterization to actual realization of these in physical systems is nevertheless impeded by a number of difficulties. Many stabilizer Hamiltonians are composed of non-local or many-body terms that are difficult to realize in practice. This usually requires significant effort and associated overhead in quantum engineering of interactions.  Both trapping of ions in Paul traps and of atoms or molecules in optical lattices offer the ability to generate the required many-body interactions. Additionally, a robust simulation of a lattice Hamiltonian requires an entropy sink to remove entropy that accumulates from noise in the control operations and interactions with the environment~\cite{Brown2007,Lloyda,Herdman2010c}. Entropy can be removed either actively via algorithmic error correction, or passively via an effective coupling to an external reservoir~\cite{Schirmer2010,Verstraete2009a,Lloyda,Kraus2008,Diehl2008}. In this work we will show how to overcome these challenges for the robust quantum simulation of both abelian and non-abelian toric code Hamiltonians and generate an effective coupling to a low temperature thermal reservoir.

In the rest of this paper we first provide an introduction to stabilizer Hamiltonians in Section~\ref{sec:stabilizer}.  In Section~\ref{sec:toric_code} we then summarize the properties of the well known abelian and non-abelian toric code Hamiltonians of Kitaev, which constitute key examples of stabilizer Hamiltonians with topologically ordered ground states and abelian and non-abelian excitations, respectively.  We then present a detailed discussion of thermalization for stabilizer Hamiltonians in Section~\ref{sec:thermalization_of_toric_code}.  The analysis given here provides both a generalization and a more efficient approach to generating thermalization than that employed in our previous work~\cite{Herdman2010c}.  This improved approach to thermalization constitutes the main new set of results in this work. Following this, we summarize several routes to simulation of the stabilizer Hamiltonian and components needed for thermalization, including a non-perturbative approach that is considerably more efficient than our previous perturbative stroboscopic approach (Section~\ref{sec:simulating_many_body_quantum_operators}).  We outline how such a thermal quantum simulation of the abelian toric code may be implemented with a finite set of neutral atoms trapped
in an optical lattice and indicate what is required in order to generalize this to thermal quantum simulation of the non-abelian toric code.  Section~\ref{sec:summary} concludes with a brief summary.

\section{Stabilizer Hamiltonians}
\label{sec:stabilizer}

Consider a class of lattice systems whose degrees of freedom are $d-$level systems (qudits). We consider Hamiltonians with local $n$-body interactions
\begin{align}
\label{eq:general_stabilizer}
H = -\sum_\nu J_\nu \sum_\mathcal{N} h_\mathcal{N}^\nu
\end{align}
where $\mathcal{N}$ is a spatial neighborhood, $\nu$ labels the type of interaction, and $\{J_\nu\}$ are the interaction strengths with $J_\nu > 0$. We assume that $\{h_\mathcal{N}^\nu\}$ are local $n$-body projection operators with maximum eigenvalue $1$. We will now consider the eigenstates of a local term $h \in \{ h_\mathcal{N}^\nu \}$. We will label the eigenstates of $h$ as $\ket{ \epsilon_d }$, where $ \epsilon > 0$, the eigenvalue of $h$ is $(1-\epsilon)$, and the label $d$ distinguishes degeneracies:
\begin{align}
h \ket{ \epsilon_d } = (1-\epsilon) \ket{ \epsilon_d }.
\end{align}
With this notation, the ground state of each local term in the Hamiltonian is $\ket{0}$, with eigenvalue $1$ and $\epsilon = 0$. We can consider the eigenoperators $\{ b_d,b_d^\dagger \}$ that span the eigenspace of $h_\mathcal{N}^\nu$:
\begin{align}
b_d^\dagger \ket{ \epsilon_c } = \delta_{\epsilon,0} \ket{ \epsilon_d }, &\quad b_d \ket{ \epsilon_c} = \delta_{d,c} \ket{0} \\
&h = \mathbb{1} - \sum_d b_d^\dagger b_d
\end{align}
An arbitrary eigenoperator can be formed from products of $\{ b_d,b_d^\dagger \}$.

We will now consider the case of such Hamiltonians for which all the $h_\mathcal{N}^\nu$ commute.  These constitute the class of stabilizer Hamiltonians, whose ground states include the well known stabilizer codes ~\cite{Bacon2006,Yoshida2011a,Dennis2002}. In this case, eigenstates of $H$ will be simultaneous eigenstates of all $h_\mathcal{N}^\nu$. In particular the ground states of $H$ will be the ground state of all local terms:
\begin{align}
h_\mathcal{N}^\nu \ket{ \Psi_0 } = \ket{\Psi_0}\quad \forall \left(\mathcal{N},\nu \right)
\end{align}
Excited eigenstates of $H$ can be generate by applying products of $b_d^\dagger$ to the ground state. For example, the state
\begin{align}
\ket{ \Psi_{ex} [ \left( \mathcal{N}, \mu, d \right), \left( \mathcal{N}', \mu, d \right) ] } = b^\dagger_{\mathcal{N},\mu,d} b^\dagger_{\mathcal{N}',\mu,d} \ket{\Psi_0}
\end{align}
has purely \emph{localized} quasiparticle excitations at neighborhoods $\mathcal{N}$ and $\mathcal{N}'$. Therefore, the eigenoperators of $H$ are consequently purely local, and can be formed from local superpositions of products of the $\{ b^\dagger_{\mathcal{N},\mu,d} \}$. This locality of the eigenoperators is a key feature of stabilizer-like Hamiltonians.  We expect that an arbitrary translationally invariant local Hamiltonian with non-communting terms will have momentum eigenstates, and therefore the eigenoperators will be extensive superpositions of local operators.

The energy cost of a local quasiparticle excitation  $\left( \mathcal{N}, \mu, a \right)$ is $\Delta E_a^\mu = \epsilon^\mu_a J_\mu$. An arbitrary one-qudit operator acting on qudit $j$, $\sigma_j$, will be formed from a finite local sum of  products of the eigenoperators of all neighborhoods $\mathcal{N}_i, i \in \mathcal{N}_i$. Therefore an arbitrary  one-qudit operator will only create, annihilate or translate quasiparticles within the local region of neighborhoods $\mathcal{N}_i$. In contrast, an arbitrary Hamiltonian with non-commuting local terms will have eigenstates with propagating quasiparticles; consequently a local operator acting on an eigenstates will generically create an non-local superposition of quasiparticles.

\section{Topological Phases and Toric Codes}
\label{sec:toric_code}
Topologically ordered phases of matter are 2D quantum liquid states with no broken conventional symmetry~\cite{Wen1990,Nayak2008}. These topological phases have a quantum ordering which cannot be detected by a local order parameter. The topological order results in a robust ground state degeneracy on surfaces with a non-trivial topology and a finite energy gap to anyonic quasiparticle excitations. Different ground states can only be distinguished by non-local operators that wind around a non-contractible loop of the surface.

Such topological phases have been proposed as the basis for a physically fault tolerant quantum computer~\cite{Freedman2002a,Kitaev2003,Nayak2008,Dennis2002}. Quantum information can be encoded in the degenerate ground states; since all local operators cause transitions to excited state above the finite gap, these phases are relatively insensitive to local perturbations.  Furthermore, both the tunneling amplitude between ground states and splitting of the ground state degeneracy are suppressed exponentially in the system size.  Logical operations can be performed by creating and braiding the anyonic excitations~\cite{Bonesteel2005}. Since the result of braiding operations depends only on the topological properties of the braids, not the precise details of the braiding paths, these logical control operations are intrinsically robust against noisy control operations. For certain phases with a sufficiently rich topological order, braiding operations form a universal set of quantum gates~\cite{Freedman2002a,Kitaev2003,Mochon2004,Mochon2003}.

To robustly physically realize such a robust topological phase, it is essential to maintain equilibrium with a low temperature external reservoir that can remove entropy and any associated accumulation of excitations due to environmental noise and/or noisy control operations~\cite{Brown2007,Kitaev2006a,Dennis2002,Nayak2008,Alicki2009a,Alicki2007a}. A generic quantum simulation (e.g., with trapped cold atoms) is an open quantum system that is not intrinsically in equilibrium with an external thermal reservoir. Thus maintaining thermal equilibrium at low temperatures requires explicitly generating an effective coupling to an external reservoir. It is important to note there that while theoretical studies have shown that in the thermodynamic limit, topological order is destroyed at any finite temperature \cite{Nussinov2008a}, in a finite sized system there is a finite temperature crossover below which the topological order is preserved \cite{Castelnovo2007a}.

Kitaev has introduced a class of exactly soluble lattice models with topologically ordered ground states~\cite{Kitaev2003,Bravyi1998,Wen2003}. The Hilbert space of these systems is defined in general by a set of qudits that sit on the links of an oriented square lattice. When this model is placed on a lattice on a torus, this is known as the toric code Hamiltonian.  We outline here the generic model of the abelian toric code as well as its non-abelian generalizations.  Each qudit state is labeled by an element of a finite group $G$, such that the local Hilbert space on each link is $\left\{ \ketil{g}, g \in G \right\}$. For each qudit, we define the operators that perform left and right multiplication
\begin{align}
L_+^h \ket{g} = \ket{hg} \quad L_-^h \ket{g} = \ket{gh^{-1}},
\end{align}
as well as the projection operators
\begin{align}
T_+^h \ket{g} = \delta_{h,g} \ket{g} \quad T_-^h \ket{g} = \delta_{h-1,g} \ket{g}.
\end{align}

The toric code Hamiltonian is a stabilizer-like Hamiltonian comprising commuting 4-body interactions:
\begin{align}
H_G = -\sum_v A_v - \sum_p B_p,
\end{align}
where the $A_v$'s are 4-body interactions defined on the vertices of the lattice and the $B_p$'s are 4-body interactions defined on the plaquettes of the lattice. The vertex terms are given by
\begin{align}
A_v = \frac{1}{\left \vert G \right \vert} \sum_{g \in G} {L_+^g}_i {L_+^g}_ j {L_-^g}_k {L_-^g}_l,
\label{eq:A_nonabelian}
\end{align}
where $\{i,j,k,l \} \in v$ and are ordered clockwise around $v$. The vertex operators can be viewed as gauge transformations, and eigenvalue $1$ eigenstate of $A_v$ are considered gauge invariant. Any state for which $A_v \ketil{\Psi_0 } \neq \ketil{ \Psi_0 }$ on a given vertex is therefore not gauge invariant, and considered to have a non-trivial electric charge at vertex $v$.

The plaquette operators are given by:
\begin{align}
B_p = \sum_{ g_1 g_2 g_3 g_4 = \mathrm{1} } {T_-^{g_1}}_i {T_-^{g_2}}_ j {T_+^{g_3}}_k {T_+^{g_4}}_l,
\label{eq:B_nonabelian}
\end{align}
where the sum over $\{ g_1,g_2,g_3,g_4 \}$ whose product is the identity. Any state which is not a $1$ eigenvalue eigenstate of some $B_p$ is considered to have a non-trivial magnetic charge on the face of plaquette $p$.

All the $A_v$ and $B_p$ commute, and so the ground state of $H_G$ is a simultaneous eigenstate of all $A_v$ and $B_p$ operators with eigenvalue 1:
\begin{align}
A_v \ket{\Psi_0 } = \ket{ \Psi_0 }, \quad B_p \ket{ \Psi_0 }  =  \ket{ \Psi_0 }.
\end{align}
The ground state has no electric and magnetic charges, and there is a finite gap to electric and magnetic quasiparticle excitations. The stabilizer-like form of $H$ means that all excited states have purely localized electric or magnetically charged quasiparticle excitations. The spectrum of $H$ includes electric charges on the vertices, magnetic charges on the plaquettes, and bound dyonic state of electric and magnetic charges.

Magnetic charges are labeled by the conjugacy classes of $G$, where the conjugacy class $C_h$ of element $h$ is defined as
\begin{align}
C_h \equiv \left \{ ghg^{-1}; g \in G \right \}.
\end{align}
The centralizers of each conjugacy class $C_h$, which commute with all elements of $C_h$ label the electric charges. Pairs of neighboring excitations can be created by applying one-qudit operators to a link. For example, to create a pair of magnetic fluxes on neighboring plaquettes, we can define an operator $E^+_{p,p'} ([h])$ which acts on the link connecting the plaquettes $p$ and $p'$~\cite{Brennen2009b}:
\begin{align}
\ket{ m_p^{[h]}, m_{p'}^{[h]} } = E_{p,p'}^+ ([h]) \ket{ \Psi_0 }, \quad E_{p,p'}^+ ([h]) \equiv \frac{1}{\sqrt{\vert [h] \vert}} \sum_{h \in [h]} L_-^h,
\end{align}
\noindent
where $\ketil{ m_p^{[h]}, m_{p'}^{[h]} }$ is a state with magnetic fluxes of charge $[h]$ on plaquettes $p,p'$. The operator $E^+_{p,p'} ([h])$ is a one qudit operator acting on the link connecting $p$ and $p'$. An arbitrary one-qudit operator applied to the ground state will crate a superposition of electric and magnetic charges at the neighboring vertices and plaquettes.

\subsection{Abelian Toric Code}
\label{subsec:abelianTC}
The simplest choice of $G=Z_2$ results in Kitaev's canonical toric code~\cite{Kitaev2003,Bravyi1998}, which has an abelian $Z_2$ topologically ordered ground state. The $Z_2$ toric code has a 4-fold degenerate ground state on the torus. Since the braiding statistics are abelian in this case, the $Z_2$ toric code can act only as a topologically protected quantum memory and not as a universal topological computer~\cite{Dennis2002,Kitaev2003}.

The Hamiltonian of the $Z_2$ toric code is given by
\begin{equation}
H_{\mathrm{TC}} = -\lambda_e \sum_v A_v - \lambda_m \sum_p B_p  \label{eq:H_TC}, \quad A_v \equiv \prod_{j \in v} \sigma^z_j, \quad B_p \equiv  \prod_{j\in p} \sigma^x_j,
\end{equation}
where $\{\sigma_j\}$ are a set of qubits located on the links of a square lattice on a torus, and $\sigma_j^\alpha$ is the $\alpha \in \{x,y,z\}$ Pauli operator on qubit $j$. $A_v$ and $B_p$ are the 4-body interactions around the vertices $v,$ and plaquettes $p$ of the lattice, respectively. The ground state of $H_{\mathrm{TC}}$ is a quantum liquid state in which all local correlation functions decay exponentially. Nevertheless, the ground state shows $Z_2$ topological order, manifested in the four-fold ground state degeneracy of $H_{\mathrm{TC}}$ on the torus. These degenerate ground states may be distinguished by the action of the following non-local loop operators:
\begin{equation}
  W^x_{1,2} \equiv \prod_{j \in c_{1,2}} \sigma^x_j, \quad W^z_{1,2} \equiv \prod_{j \in \tilde{c}_{1,2}} \sigma^z_j.
\end{equation}
Here $c_{1,2}$ are loops through the vertex lattice that wind around one of the two directions of the torus, and $\tilde{c}_{1,2}$ are loops passing through the faces of the plaquettes. Since $[ W^{x,z}_{1,2}, H_{\mathrm{TC}} ] = 0$, the ground states may also be written as eigenstates of $W^{x,z}_{1,2}$ with eigenvalues $\pm1$.

The excited eigenstates with $A_v \ket{\Psi_{ex}} = - \ket{\Psi_{ex}}$ have a localized ``electric" $e$-type quasiparticles on the vertex $v$ that costs energy $2 \lambda_e$.  Eigenstates with $B_p \ket{\Psi_{ex}} = - \ket{\Psi_{ex}}$ have a localized ``magnetic" $m$-type quasiparticles on the plaquette $p$ that costs energy $2 \lambda_m$. Pairs of quasiparticles can be created by applying one-body Pauli operators to the ground state:
\begin{align}
\ket{e_v e_{v'}} = \sigma^x_{v,v'} \ket{\Psi_0^{\mathrm{TC}}}, \quad \ket{m_p m_{p'}} = \sigma^z_{p,p'} \ket{\Psi_0^{\mathrm{TC}}}
\end{align}
where $\sigma_{v,v'}$ is spin on the link connecting $v$ and $v'$ and $\sigma_{p,p'}$ connects plaquettes $p$ and $p'$. The lowest energy excited states are characterized by such pairs of localized quasiparticles and are separated from the ground state by an energy gap of  $\Delta_{e,m} = 4\lambda_{e,m}$. An arbitrary one-body operator acting on an eigenstate of $H_{\mathrm{TC}}$ will act to either create or destroy pairs of quasiparticles around the link: sequential action of one-body operators therefore results in translation of a single quasiparticle across the link. Both $e$ and $m$ quasiparticles act as bosons under exchange amongst their own type: however, braiding an $e$ around an $m$ generates a phase of $-1$, and so the two types of quasiparticles are seen to have mutual abelian \emph{semionic} braiding statistics.

On a finite-sized lattice of linear dimension $L$, the crossover temperature below which the topological order is preserved for the $Z_2$ toric code is given by $T^* \sim \Delta/ \ln{L}$ \cite{Castelnovo2007a}. The anyonic braiding statistics allow this abelian toric code to therefore act as the basis for a robust quantum memory as long as it is kept in equilibrium at a low temperature, $T \ll T^*$.  However, as noted above, the abelian topologically protected braiding operations are not universal for computation in the $Z_2$ model.

\subsection{Nonabelian Toric Code}
\label{sub:nonabelian_toric_code}

Mochon \cite{Mochon2003,Mochon2004} has shown that for certain non-abelian finite groups $G$, the braiding and fusion of electric and magnetic fluxes can lead to formation of a universal logical gate set. Thus, if the corresponding toric code Hamiltonians $H_G$ can be robustly generated and localized quasiparticles controllably created and manipulated, the non-Abelian toric code may act as the basis for a topologically fault tolerant quantum computer. The smallest non-abelian group is the group of permutations of three objects, $S_3$ and this would already allow for a topologically protected universal gate set~\cite{Mochon2003,Mochon2004}. Since $S_3$ has 6 elements, simulating $H_{S_3}$ therefore requires using qudits with $d=6$ at each link of the square lattice. In Section~\ref{sec:simulating_many_body_quantum_operators} we indicate how the corresponding vertex and plaquette operators of Eqs.~(\ref{eq:A_nonabelian})-(\ref{eq:B_nonabelian}) may be constructed from a universal set of one and two-qudit gates.

\subsection{Finite Temperature Behavior}

As noted above, while topological order is unstable at any finite temperature in the thermodynamic limit, in a finite sized system there is a finite crossover temperature below which the topological order may be preserved~\cite{Castelnovo2007a,Nussinov2008a}. To keep the system in such effective low temperature state with respect to $H_G$, or to cool all the way to the ground state, we must therefore couple each vertex and plaquette to a set of ancillary reservoir qudits undergoing dissipation~\cite{Alicki2009a,Alicki2007a}.  In the next section we describe how this thermalization may be carried out in an efficient manner, focusing on the qubit case, i.e., on the abelian $Z_2$ toric code.

\section{Thermalization of stabilizer Hamiltonians}
\label{sec:thermalization_of_toric_code}

Since noisy control operations and interactions with the environment will introduce entropy to the system, a robust quantum simulation requires a dissipative process to remove entropy and effectively cool the system~\cite{Brown2007,Lloyda}. As an alternative to algorithmic error correction, we present an approach here to maintain the system in equilibrium with an effective thermal reservoir. If the effective temperature of the reservoir approaches zero, this thermalization will act purely as passive error correction; however, our approach also provides for the ability to tune the temperature of the reservoir and equilibrate the system at finite temperature. In addition to providing robustness, this thermal equilibration also therefore would allow for a study of thermal properties of the quantum model at hand.
One procedure for extracting finite temperature properties of a many-body system is to simulate its thermalization by coupling it to external dissipative modes. The steady-state of the following Lindblad master equation is guaranteed to produce the thermal equilibrium density matrix of the system:
\begin{equation}
\frac{\dt \rho}{\dt{t}} = -\frac{i}{\hbar}[H_0,\rho] + \sum_k \gamma_k \mathcal{D}[H^-_k] \rho + \sum_k e^{-\beta \hbar \omega_k}\gamma_k \mathcal{D}[H^+_k]\rho
\label{eq:thermal_me}
\end{equation}
where $\mathcal{D}[A]\rho \equiv 2 A \rho A^\dagger - A^\dagger A\rho - \rho A^\dagger $, and $H^+_k$ creates an excitation of energy $\hbar\omega_k$. The unique steady state of this master equation is the thermal density matrix corresponding to the inverse temperature $\beta$ and Hamiltonian $H_0$ \cite{Breuer2002}. Despite this simple description, the mathematical model above can be difficult to simulate in practice because the excitation creation and annihilation operators $H^+_k, H^-_k$ are usually a superposition of non-local many-body operators in an interacting system. For the abelian toric code these are $4$-body operators, and therefore each generator in the Lindblad terms is a $4$-body term.

These $4$-body operators can be implemented using the stroboscopic technique described in Ref. \cite{Herdman2010c}.  However such an implementation is resource intensive.  Here we describe a simpler implementation of thermalization in stabilizer Hamiltonians by exploiting the local properties of excitations. Our analysis in this section focuses on stabilizer Hamiltonians of qubits for simplicity, however the arguments presented here may be extended to qudit Hamiltonians, as required for realization of, e.g., a non-abelian toric code.  The key insight for constructing a simple thermalization scheme is to note that a local Pauli operation in the interaction picture defined by the stabilizer Hamiltonian (an eigenoperator decomposition of the local Pauli) decomposes into a small number of Fourier components:
\begin{equation}
	e^{\frac{i}{\hbar}t H^\textrm{stab}} \sigma^{\alpha}_j e^{-\frac{i}{\hbar}t H^\textrm{stab}}
		= \sum_{k=1}^M e^{-\frac{i}{\hbar}2\epsilon_{\alpha,j,k} t}a_{\alpha,j,k} + e^{\frac{i}{\hbar}2\epsilon_{\alpha,j,k} t}a^\dagger_{\alpha,j,k}
\label{eq:fourier_general}
\end{equation}
where $\alpha \in \{x,y,z\}$, $j$ indexes a qubit in the many-body system, and $a_{\alpha,j,k}, a_{\alpha,j,k}^\dagger$ are generalized creation and annihilation operators for excitations of energy $\epsilon_{\alpha,j,k}$, associated with the Pauli operator $\sigma^\alpha_j$. While for a general Hamiltonian this eigenoperator decomposition can have a number of terms that is extensive in system size, the number of terms above, $M$, will necessarily be small because the terms in $H_\textrm{stab}$ commute with themselves and only a small number of them do not commute with the local Pauli operator $\sigma^\alpha_j$.

In our construction we will interact each local qubit with up to $M$ ancillary spins that are driven to a thermal state. The specific form of the interaction Hamiltonian is:
\begin{equation}
	H_\textrm{int}
		= \sum_{j=1}^N \sum_{\alpha} \sum_{k=1}^M \sigma^\alpha_j \otimes \Sigma^x_{\alpha,j,k}
\end{equation}
where $\Sigma^x_{\alpha,j,k}$ is the Pauli $x$ operator on ancilla spin $k$ for the $\alpha$th Pauli operator on qubit $j$ in the stabilizer Hamiltonian. Note that this interaction Hamiltonian is two-body and each lattice site interacts with at most $M$ ancillas.. The Zeeman frequency of the $k$th ancilla spin is chosen to be $\omega_{\alpha,j,k} = \epsilon_{\alpha,j,k}/\hbar$.

The effective dynamics of the system and ancillary spins, with combined density matrix $\Omega \in \mathcal{H}^{\textrm{lattice}}\otimes \left(\mathbb{C}^2\right)^{\otimes MN}$, is given by:
\begin{align}
	\frac{\dt{\Omega}}{\dt{t}}
		&= -\frac{i}{\hbar}[H^\textrm{stab}
			- \sum_{\alpha, k, j} \hbar\omega_{\alpha,j,k} \Sigma^{z}_{\alpha,j,k}
		         + H_\textrm{int}, \Omega] \nonumber \\
		& + \sum_{\alpha,j,k} \gamma_{\alpha,j,k}^-\mathcal{D}[\Sigma^{-}_{\alpha,j,k}]\Omega
			+ \sum_{\alpha,j,k} \gamma_{\alpha,j,k}^+ \mathcal{D}[\Sigma^{+}_{\alpha,j,k}]\Omega
\end{align}
The dissipation rates for all the ancillary spins are chosen such that
\begin{equation}
	\frac{\gamma^-_{\alpha,j,k}}{\gamma^+_{\alpha,j,k}}
		= e^{-\beta \hbar \omega_{\alpha,j,k}}
\end{equation}
so that in the absence of the coupling to stabilizer Hamiltonian qubits, the unique steady state of dissipative dynamics of a single ancillary spin is the thermal density matrix at inverse temperate $\beta$:
\begin{equation}
\left(\begin{array}{cc}p^0_{\alpha,j,k} & 0 \\0 & p^1_{\alpha,j,k}\end{array}\right)
\end{equation}
with $p^1_{\alpha,j,k} = 1-p^0_{\alpha,j,k} = e^{-\beta \hbar \omega_{\alpha,j,k}}p^0_{\alpha,j,k}$.
Such a thermally driven ancilla system can be implemented as a driven three-level lambda-configuration atom with a strong spontaneous emission channel, resulting in the two stable levels having thermal population distribution; in such a case they are pseudospins. The inverse temperature of the ancilla pseudospins is chosen to be $\beta = 1/k_BT$ (where $T$ is the desired temperature for the simulation).

Transforming into the interaction picture with respect to $H^\textrm{stab} - \sum_{\alpha,j,k}  \hbar\omega_{\alpha,j,k}  \Sigma^{z}_{\alpha,j,k}$, and using \erf{eq:fourier_general} yields an interaction picture interaction Hamiltonian of the form:
\begin{align}
\tilde{H}_\textrm{int}
		&= \sum_{j=1}^N \sum_{\alpha} \sum_{k=1}^M \left( \sum_{l=1}^M e^{-\frac{i}{\hbar}2\epsilon_{\alpha,j,l} t}a_{\alpha,j,l} + e^{\frac{i}{\hbar}2\epsilon_{\alpha,j,l} t}a^\dagger_{\alpha,j,l} \right) \otimes \left( \Sigma^-_{\alpha,j,k}e^{-i2\omega_{\alpha,j,k}t} +  \Sigma^+_{\alpha,j,k}e^{i2\omega_{\alpha,j,k}t} \right)
\end{align}
Remembering that  $\hbar\omega_{\alpha,j,k} = \epsilon_{\alpha,j,k}$ and using the rotating wave approximation to drop all oscillating terms reduces the interaction Hamiltonian to
\begin{equation}
\tilde{H}^{\textrm{RWA}}_\textrm{int} =\sum_{j=1}^N \sum_{\alpha} \sum_{k=1}^M a_{\alpha,j,k}\otimes\Sigma^+_{\alpha,j,k} + a^\dagger_{\alpha,j,k} \otimes \Sigma^-_{\alpha,j,k}
\end{equation}
Therefore the effective evolution in the interaction picture and under the rotating wave approximation is given by the master equation:
\begin{align}
	\frac{\dt{\Omega}_\textrm{int}}{\dt{t}}
		&= -\frac{i}{\hbar}[\sum_{j=1}^N \sum_{\alpha} \sum_{k=1}^M a_{\alpha,j,k}\otimes\Sigma^+_{\alpha,j,k} + a^\dagger_{\alpha,j,k} \otimes \Sigma^-_{\alpha,j,k}, \Omega] \nonumber \\
		& + \sum_{\alpha,j,k} \gamma^-_{\alpha,j,k}\mathcal{D}[\Sigma^{-}_{\alpha,j,k}]\Omega
			+ \sum_{\alpha,j,k} \gamma^+_{\alpha,j,k} \mathcal{D}[\Sigma^{+}_{\alpha,j,k}]\Omega
\label{eq:me_rwa1}
\end{align}
This evolution describes energy exchange of the stabilizer Hamiltonian system with a set of thermalized ancilla spins, and will result in the thermalization of the system. Essentially the engineered dissipation channels provided by the ancilla pseudospins mimic a bath satisfying detailed balance at the resonance frequencies of the system. Although we can demonstrate the thermalization of the system in this general setting (see Appendix), for concreteness we will now illustrate it explicitly for the abelian toric code.

\begin{figure}
\begin{align*}
	E^\dagger_{j,\mathcal{E}} \ket{\myvcenter{\includegraphics[width=10ex]{empty.png}}}
		&= \ket{\myvcenter{\includegraphics[width=10ex]{create_m.png}}} \\
	T_{j,\mathcal{E}} \ket{\myvcenter{\includegraphics[width=10ex]{up.png}}}
		&= \ket{\myvcenter{\includegraphics[width=10ex]{dn.png}}}\\
	E^\dagger_{j,\mathcal{M}} \ket{\myvcenter{\includegraphics[width=10ex]{empty.png}}}
		&= \ket{\myvcenter{\includegraphics[width=10ex]{create_e.png}}} \\
	T_{j,\mathcal{M}} \ket{\myvcenter{\includegraphics[width=10ex]{left.png}}}
		&= \ket{\myvcenter{\includegraphics[width=10ex]{right.png}}}
\end{align*}
\caption{Graphical representation of the action of excitation creation and translation operators for the toric code. The black dot indicates site (qubit) $j$, the blue (red) circles indicate electric (magnetic) excitations. \label{fig:operator_action}}
\end{figure}

For the toric code, the Fourier decomposition of the local Pauli terms of interest is:
\begin{equation}
	e^{\frac{i}{\hbar}t H_\textrm{TC}} \sigma^{x/z}_j e^{-\frac{i}{\hbar}t H_\textrm{TC}}
		= e^{-2\frac{i}{\hbar}\Delta t}E_{j,\mathcal{E}/\mathcal{M}}
		+ e^{2\frac{i}{\hbar}\Delta t}E_{j,\mathcal{E}/\mathcal{M}}^\dagger
		+ T_{j,\mathcal{E}/\mathcal{M}}
\label{eq:localexc_toric}
\end{equation}
$\nu=\mathcal{E}$ and $\nu=\mathcal{M}$ denote electric and magnetic excitations, and we assume the electric and magnetic excitations have the same energy: $\lambda_e = \lambda_m = \Delta$. The operator $E^\dagger_{j,\nu}$ creates a pair of excitations of type $\nu$ about site $j$ and $T_{j,\nu} = T^\dagger_{j,\nu}$ translates a type-$\nu$ excitation about site $j$. See Ref. \cite{Herdman2010c} for the formal definition of these operators, and see Fig. \ref{fig:operator_action} for graphical representation of the action of these operators.  In terms of these operators, the toric code Hamiltonian may be written as,
\begin{equation}
\label{eq:HTCfourier}
	H_{\rm TC} = \frac{\Delta}{8} \sum_{j,\nu} \left(2E_{j,\nu}^\dagger E_{j,\nu} + T_{j,\nu}^2\right)
\end{equation}

This decomposition of toric code excitations was used by Alicki, Fannes, and Horodecki \cite{Alicki2009a,Alicki2007a} to show that thermalization of the toric code is guaranteed under a local interaction Hamiltonian coupling to a thermal bath of the form:
\begin{equation}
	H_\textrm{int,harmonic}
		= \sum_{j=1}^N \sigma^x_j \otimes f_j + \sum_{1}^N \sigma^z_j \otimes g_j
\end{equation}
where $f_j, g_j$ are Hermitian bath operators that satisfy detailed balance. For example, in the case of a bath of free harmonic modes, $f_j, g_j$ could be the sum over displacement operators for the modes. We will instead, show that by the above construction one can also simulate thermalization by utilizing ancillary pseudospins as the engineered dissipative environment. The interaction Hamiltonian we require for the toric code example is
\begin{equation}
	H_\textrm{int}
		= \sum_{j=1}^N \sigma^x_j \otimes  \Sigma^x_{\Delta, j, \mathcal{E}} + \sigma^z_j \otimes  \Sigma^x_{\Delta, j, \mathcal{M}}  + \sigma^x_j \otimes  \Sigma^x_{0, j, \mathcal{E}} + \sigma^z_j \otimes  \Sigma^x_{0, j, \mathcal{M}}
\end{equation}
Here, the $\Sigma^x_{\Delta/0, j, \mathcal{E/M}}$ are Pauli-$x$ operators on ancillary pseudospins which are implemented as driven three-level lambda-configuration atoms with strong spontaneous emission. Each toric code lattice spin has four types of ancillary pseudospin coupled to it ($\Sigma_{\Delta,\mathcal{E}}, \Sigma_{0, \mathcal{E}}, \Sigma_{\Delta, \mathcal{M}}$ and $\Sigma_{0, \mathcal{M}}$). The characteristic frequency of all $\Delta$-type ancillary pseudospins is $\omega_\Delta = \Delta/\hbar$, and all $0$-type ancillary pseudospins is $\omega_0 = 0$ (these correspond to the two eigenfrequencies in the decomposition \erf{eq:localexc_toric}). Transforming into the interaction picture with respect to $H_\textrm{TC} - \sum_{j,\nu} \hbar\omega_\Delta \Sigma^{z}_{\Delta, j, \nu} - \sum_{j} \hbar\omega_0 \Sigma^{z}_{0, j, \nu}$, with $\nu \in \{\mathcal{E}, \mathcal{M} \}$, and employing the rotating wave approximation produces:
\begin{equation}
\tilde{H}^{\textrm{RWA}}_\textrm{int} = \sum_{j,\nu} E_{j,\nu}\otimes \Sigma^+_{\Delta, j, \nu} + E_{j,\nu}^\dagger \otimes \Sigma^-_{\Delta, j, \nu} +  T_{j,\nu} \otimes \Sigma^x_{0, j, \nu}
\end{equation}

Therefore the effective evolution in the interaction picture and under the rotating wave approximation is given by the master equation:
\begin{align}
	\frac{\dt{\Omega}}{\dt{t}}
		&= -\frac{i}{\hbar}[\sum_{j,\nu} E_{j,\nu}\otimes \Sigma^+_{\Delta, j, \nu} + E_{j,\nu}^\dagger \otimes \Sigma^-_{\Delta, j, \nu} +  T_{j,\nu} \otimes \Sigma^x_{0, j, \nu}, \Omega] \nonumber \\
		& + \sum_{j,\nu} \gamma^-_\Delta \mathcal{D}[\Sigma^{-}_{\Delta, j, \nu}]\Omega
			+ \sum_{j} \gamma^+_\Delta \mathcal{D}[\Sigma^{+}_{\Delta, j, \nu}]\Omega
			+ \sum_{j} \gamma^-_0 \mathcal{D}[\Sigma^{-}_{0, j, \nu}]\Omega
			+ \sum_{j} \gamma^+_0 \mathcal{D}[\Sigma^{+}_{0, j, \nu}]\Omega
			\equiv \mathcal{L}_\textrm{eff}\Omega
\label{eq:me_rwa2}
\end{align}
If we choose $\gamma^+_\Delta = e^{-\beta_\mathsf{T}\Delta}\gamma^-_\Delta$ and $\gamma^+_0 = \gamma^-_0$, which results in the $\Delta$-type ancillary pseudospins thermalizing to a finite inverse temperature $\beta$ and the $0$-type ancillary pseudospins thermalizing to infinite temperature ($\beta=0$), it is shown in the Appendix that the unique steady state of this master equation is the thermal state of the combined system at inverse temperature $\beta$. That is,
\begin{equation}
\mathcal{L}_\textrm{eff} ~\frac{e^{-\beta H_\textrm{TC}}\otimes e^{-\beta \left( - \sum_{j,\nu} \hbar\omega_\Delta \Sigma^{z}_{\Delta, j, \nu} - \sum_{j,\nu} \hbar\omega_0 \Sigma^{z}_{0, j, \nu}\right)}}{\mathcal{Z}} = 0
\label{eq:toric_ss}
\end{equation}
and this is the only state that satisfies this property.

The above analysis for simulating thermalization focused explicitly on qubit stabilizer Hamiltonians. However, the same construction follows for qudit stabilizer Hamiltonians because the critical property of local excitations and a decomposition analogous to \erf{eq:fourier_general} also holds for these. In the qudit case however, there are more than three Pauli generators in the group of local generators and thus the Fourier decomposition will be more involved. As a consequence the number of ancilla spins necessary to simulate the thermal bath will also increase.

\section{Physical simulation of stabilizer system}
\label{sec:simulating_many_body_quantum_operators}

We shall now consider the physical context in which the aforementioned thermal stabilizer system is to be simulated.  Though a number of proposals exist for quantum simulation, for specificity we shall restrict our discussion to arrays of trapped neutral atoms.  In particular, we consider a set of $\sim 250$ individual $^{133}$Cs atoms trapped at the sites of an addressable, simple cubic optical lattice~\cite{Nelson2007a}.   The orbital degrees of freedom are slow, and can be considered as effectively frozen on the time scales relevant to our analysis.  We therefore need consider only the internal atomic degrees of freedom, from which we select two hyperfine levels (e.g., $\ket{F,m_F} = \ket{4,4}, \ket{3,3} $) to define a 2-level pseudospin system.  The Hamiltonian, \erf{eq:H_TC}, will be implemented in an interaction picture with respect to the atomic energy levels. Additionally, we choose auxiliary internal levels to serve as intermediate states to facilitate optical frequency Raman transitions for single qubit operations, as well as a highly excited, $n\simeq 80$, Rydberg level necessary for two qubit interactions~\cite{Jaksch2000, Cozzini2006}.  Since the pseudospins are localized at the sites of a cubic lattice, one can choose to either realize the dynamics on a single plane using a surface code ~\cite{Bravyi1998, Freedman2001} or in a three-dimensional cubic array with toroidal boundary conditions realized by SWAP operations.   In addition to this set of system qubits on which the stabilizer Hamiltonian is simulated, we must include a set of ancillary atoms to serve as a thermal reservoir.  This ancillary set must be strongly dissipative and will be optically pumped to produce the desired thermal state.

The analysis of the previous section demonstrates that the thermal properties of stabilizer Hamiltonians may be studied by coupling to a dissipative bath of two-level systems.  However, it is not possibly to directly implement the Liouvillian on the neutral atom system, so we instead take a stroboscopic approach.  This approach applies a sequence of local operations to generate an evolution which closely approximates the evolution generated by the dynamical equations.  We shall begin by considering the generic Lindblad master equation,
\begin{equation}
\label{eq:liouvmaster}
	\dot\rho = \mathcal{L}\rho(t) = \sum_j \mathcal{A}[H_j]\rho(t) + \sum_k\gamma_k\mathcal{D}[K_k] \rho(t).
\end{equation}
Here we have used
	$A[H_j]\rho(t) = -i [H_j, \rho(t)]/\hbar$
to represent the adjoint action of $H_j$.  Evolution under this equation generates the time evolution operator,
\begin{equation}
\label{eq:Loft}
	L(t) = \exp(\mathcal{L} t)
\end{equation}
The master equation, \erf{eq:liouvmaster}, is impossible to implement continuously in the neutral atom array, so we approximate the evolution by a sequence of local operators using a Trotter expansion,
\begin{equation}
\label{eq:trotter}
	L(t) = \exp\left(\order{1/N^2}\right)\left(\prod_j\exp\left(\mathcal{A}[H_j] \Delta t\right)\prod_k\exp\left(\gamma_k\mathcal{D}[K_k] \Delta t\right)\right)^N,
\end{equation}
where $\Delta t = t/N$.By choosing a sufficiently large $N$, the error term can be made arbitrarily small.  Each term in the master equation may then be simulated independently over the short time $\Delta t$.  For a generic stabilizer Hamiltonians, however, many of these terms will be multibody.  Couplings derived from first principles physical interactions, on the other hand, are intrinsically two-body.  Exceptions to this are rare, and usually derive from an implicit averaging over time or intermediate degrees of freedom \cite{Weimer2010}. In Ref.~\cite{Herdman2010c}, we presented a perturbative method based on the Magnus expansion for simulating many-body interactions between qubits by application of a sequence of two-body quantum gates.  Here we present a non-perturbative method for the simulation of arbitrary $n$-body qubit gates based on a construction in Ref.~\cite{mikeandike, Barenco1995}.  This method displays significantly lower overhead in terms of total gate count as compared to the Magnus expansion approach. While the method is capable of simulating a generic product of Pauli matrices, we shall explicitly consider here only the terms necessary for simulation of the abelian toric code Hamiltonian \erf{eq:H_TC}.  The Trotter expansion leaves us to simulate unitary evolution operators of the form $U_Z(t) = \exp(-i \phi \sigma_z\sigma_z\sigma_z\sigma_z)$, with $\phi = \lambda \Delta t$. This evolution may be simulated by the sequence of CPHASE and one-qubit gates shown in Fig.~\ref{fig:circuit}.  This circuit may be readily extended by single qubit operations to achieve any product of four Pauli matrices.  For example, pre- and post- multiplying by Hadamard transformations produces $U_X = \exp(-i\phi \sigma_x\sigma_x\sigma_x\sigma_x) = H^{\otimes 4} U_Z H^{\otimes 4}$.

Simulation of the dissipative terms in \erf{eq:liouvmaster} may be effected through the use of atomic states with strong spontaneous emission.  This dissipation forms the entropy reducing part of the map.  As shown in the Appendix, the stationary state of the evolution is thermal regardless of the magnitude of the dissipation couplings, $\gamma_k$, so long as they are positive.  We may therefore choose them to be arbitrarily large, effectively replacing the operators $\prod_k\exp(\mathcal{D}[K_k] \delta_t)$ by a reset to the thermal state.  This could be accomplished in a number of ways.  For example, correctly tuned coherent driving to a state with strong dissipation can optically pump the atom directly to the thermal state.  Alternatively, one may measure the ancilla bit in the computational basis and apply a $\pi$-pulse conditioned on the result of the measurement and a classical, Boltzmann-weighted random bit.

\begin{figure}[ht]
\begin{center}
\label{fig:circuit}
\includegraphics[width=.8\columnwidth]{zzzz.pdf}
\caption{Ancilla-free quantum circuit nonperturbatively implementing action of many body Hamiltonian.  The circuit on the left utilizes gates available to the neutral atom array described in the text.  Vertical lines indicate Rydberg CPHASE gates between qubits indicated by black dots and $H$ indicates a Hadamard rotation.  Action of Hamiltonian proportional to $\sigma_x\sigma_x\sigma_x\sigma_x$ may be implemented by conjugation of the above circuit with Hadamard operations.}
\end{center}
\end{figure}

For the non-abelian toric code we need to generate the generalized 4-body interactions in \erf{eq:A_nonabelian}.  This requires an implementation of the $S^{3}$ multiplication table on $\mathbb{C}^{6}$, the single qudit Hilbert space. Given an implementation of one- and two-qudit gates, we can generate the required 4-body terms by methods similar to those above for the qubit case.  Brennen \emph{et al} \cite{Brennen2005} have demonstrated the existence of universal sets of one and two qudit gates. It remains to construct explicit instances of gate sequences. Arbitrary one-qudit gates may be implemented through entirely local unitary actions on a six level qudit pseudospin system. In the case of trapped neutral atoms, this is possible with methods similar to the single qubit unitaries.  For a pair of levels within one six level system and a corresponding pair in one of its neighbors, we may construct a Rydberg blockade, analogous to the qubit case.  This allows the explicit construction of two qudit gates. Details of this will be discussed in a forthcoming paper.

\section{Summary}
\label{sec:summary}
Motivated by the need to incorporate dissipative processes to remove entropy and provide cooling during quantum simulations, we have developed a scheme for efficient finite temperature quantum simulation of general stabilizer Hamiltonians.  These Hamiltonians are typically characterized by non-local or many-body interactions that are hard to realize.  The well known toric code Hamiltonian of Kitaev, which allows both abelian representations with qubits and non-abelian representations with qudits, is taken here as a canonical example of stabilizer Hamiltonian in order to demonstrate the approach. Our method relies on coupling of each physical qubit or qudit involved in the Hamiltonian simulation to a small number of ancillary pseudospins that are dissipatively driven to reach a specific temperature.  By using a Fourier decomposition of local Pauli operations on the physical qubits, we show that we can achieve thermalization by employing only two-body couplings between the physical qubits with a small number of ancillary pseudospins.  This is a significant improvement over our earlier work ~\cite{Herdman2010c}, which required that the dissipatively driven ancillas be coupled to the physical qubits by similar many-body interactions as those contained in the stabilizer Hamiltonians.  We illustrated the thermalization approach explicitly for the abelian toric code of Kitaev, where two-body interactions with two dissipatively driven ancilla pseudospins are all that is required to achieve thermalization of a finite set of qubits evolving under the toric code Hamiltonian. This considerably simplifies the physical implementation with neutral atoms trapped in optical lattices, for which we also presented an improved approach to quantum simulation of the toric code Hamiltonian.  The approach can readily be extended to thermalization of quantum simulations with general stabilizer Hamiltonians, in particular to the simulation of the non-abelian toric code Hamiltonian, for which the smallest qudit dimension is six.  Detailed analysis of such non-abelian simulations with trapped neutral atoms will be presented elsewhere.

\section{Acknowledgments}
\label{sec:acknowledgments}
This material is based upon work supported by DARPA under Award No. 3854-UCB-AFOSR-0041.  Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

\subsection*{Appendix A: Thermal state fixed by evolution}
\label{ap:fixed}
Here we will show that the thermal state of the abelian toric code is the unique fixed point of the engineered dissipative evolution detailed in section \ref{sec:thermalization_of_toric_code}. Consider in particular, the evolution prescribed by \erf{eq:me_rwa2}. The dissipative dynamics for the ancillary pseudospins is particularly simple since it is incoherent excitation at rate $\gamma^+$ and damping at rate $\gamma^-$ of each pseudospin independently. The \textit{only} steady-state of this evolution is the mixed state of each pseudospin
\begin{equation}
\left(\begin{array}{cc}p^0_{\Delta/0} & 0 \\0 & 1-p^0_{\Delta/0}\end{array}\right)
\end{equation}
with $p^0_{\Delta/0} = \gamma^-_{\Delta/0} / (\gamma^-_{\Delta/0} + \gamma^+_{\Delta/0})$. With the choice of rates given in the main text, this is a thermal state at inverse temperature $\beta$ for $\Delta$-type pseudospins and a completely mixed state for the $0$-type pseudospins -- i.e. $p^1_{\Delta/0}/p^0_{\Delta/0} = e^{-\beta \hbar \omega_{{\Delta/0}}}$, where $p^1_{\Delta/0} \equiv 1- p^0_{\Delta/0}$.

Hence the steady state of master equation evolution must have the form:
\begin{equation}
	\Omega_\textrm{ss}
		= \rho
		\otimes
		\underbrace{\rho_{\Delta}^{\otimes 2N}}_{\Sigma_{\Delta}
		\textrm{pseudospins}}
		\otimes
		\underbrace{\rho_{0}^{\otimes 2N}}_{\Sigma_{0} \textrm{pseudospins}}
\end{equation}
with additional conditions on $\rho$, the state of the toric code lattice spins. Here,
\begin{equation}
\rho_{\Delta} \equiv \left(\begin{array}{cc} \frac{1}{1+e^{-\beta \Delta}} & 0 \\0 & \frac{1}{1+e^{\beta \Delta}}\end{array}\right) ~~~~~ \textrm{and} ~~~~~ \rho_{0} \equiv \left(\begin{array}{cc} \frac{1}{2} & 0 \\0 & \frac{1}{2}\end{array}\right)
\end{equation}
Substituting this form into the master equation results in:
\begin{equation}
\frac{\dt{\Omega_{\textrm{ss}}}}{\dt{t}} = -\frac{i}{\hbar}\sum_{j,\nu}\left[ E_{j,\nu}\otimes \Sigma^+_{\Delta, j, \nu} + E_{j,\nu}^\dagger \otimes \Sigma^-_{\Delta, j, \nu} +  T_{j,\nu} \otimes \Sigma^x_{0, j, \nu}, \rho\otimes \rho^{\otimes 2N}_{\Delta} \otimes \rho_{0}^{\otimes 2N}\right]
\end{equation}
since the dissipative terms all evaluate to zero. Now, expanding out the steady-state form for the ancilla pseudospins, and requiring that this time derivative be zero results in the following conditions $\forall ~j, \nu$:
\begin{align}
	\label{eq:lowering}
	p^0_{\Delta} E_{j,\nu}\rho - p^1_{\Delta} \rho E_{j,\nu} &= 0\\
	\label{eq:raising}
	p^1_{\Delta} E^\dagger_{j,\nu}\rho - p^0_{\Delta} \rho E^\dagger_{j,\nu} &= 0\\
	\label{eq:translation}
	T_{j,\nu}\rho - \rho T_{j,\nu} &= 0
\end{align}

At this point note that the translation operators $T_{j,\nu}$ generate a group whose action commutes with the Hamiltonian, $H_\textrm{TC}$, and is ergodic in each energy eigenspace, i.e., any two degenerate energy eigenstates are connected by a product of $T_{j,\nu}$ (with the exception of the groundspace, which we shall address later).  Because the translation operators commute with the Hamiltonian and are ergodic on each energy eigenspace, they may be decomposed into irreducible representations as
\begin{equation}
\label{eq:irrepT}
	T_{j,\nu} = \sum_n P_n T_{j,\nu} P_n
\end{equation}
where $P_n$ is the projector onto the $n^{\rm th}$ energy eigenspace. Furthermore,
\begin{equation}
\label{eq:rhodecompose}
	\rho = \sum_{n,m} P_n \rho P_m
\end{equation}
The commutation relationship $[T_{i,\nu}, \rho] = 0$ implies then that,
\begin{align}
	P_m [T_{i,\nu}, \rho] P_n
		\notag
		&= P_m (T_{i,\nu} \rho - \rho T_{i,\nu} )P_n\\
		\label{eq:schurs}
		&= \left(P_m T_{i,\nu} P_m \right)\left(P_m \rho P_n\right) - \left(P_m \rho P_n \right)\left(P_n T_{i,\nu} P_n\right)
\end{align}
Choosing $m=n$, we see that $P_m \rho P_m$ commutes with all elements of the $m^{\rm th}$ irreducible representation of $T$.  By Schur's first lemma~\cite{Fulton1991}, this implies that $P_m \rho P_m$ must be proportional to the identity.  If we choose $m\ne n$, \erf{eq:schurs} and Schur's second lemma~\cite{Fulton1991} imply that $P_m \rho P_n = 0$.

For the groundspace, we examine the commutation relations with the string operators.  The string operators can be represented as exciton pair creations, translations and annihilations.  But the ``commutation" relations \erf{eq:lowering}-\erf{eq:translation} imply that $\rho$ commutes with any product of $T$, $E$, and $E^\dagger$ for which there are equal numbers of $E$'s and $E^\dagger$'s.  Because $P_0 \rho P_0$ commutes with all of the string operators, Schur's lemma again implies that it too must be proportional to the identity.

That the populations must satisfy the Boltzman distribution is insured by \erf{eq:lowering}, and so
\begin{equation}
\label{eq:rhothermal}
	\mathcal{L}\rho=0 \implies \rho = \frac{\exp(-\beta H_\textrm{TC})}{{\rm Tr\thinspace} \exp(-\beta H_\textrm{TC})} \otimes \rho^{\otimes 2N}_{\Delta} \otimes \rho_{0}^{\otimes 2N}
\end{equation}

Given that the thermal state is the unique steady state of this Lindblad evolution, it can also be shown that it is an attractor, meaning that all states converge to it asymptotically \cite{Schirmer2010}.
In fact, the above is an explicit demonstration of a very general statement about semigroups to be found in the work of Arveson \cite{Arveson1997} -- roughly, if a semigroup dynamics (e.g. generated by a Lindblad master equation) has an invariant state, and is ergodic, then it is the unique invariant state and is furthermore an attractor. The ergodicity of the dynamics is the key element, and for a stabilizer Hamiltonian it can be shown that if the Lindblad generators are excitation creation, annihilation and translation operators the system is ergodic. Such an argument can be used to prove that in the general qudit stabilizer Hamiltonian case a construction analogous to the one in Section \ref{sec:thermalization_of_toric_code} will fix the thermal state, and only the thermal state, of the system. This reflects a general pattern in the thermalization of stabilizer codes which we will discuss in a forthcoming paper.


text:
Find Yourself a Mentor or Two
Capt. Dallas Meachum
A lieutenant colonel provided me with important advice on mentorship when I graduated and earned my commission from Officer Candidate School at Fort Benning, Georgia. He was set to retire, but his message was clear and concise: "Find a mentor. I cannot do it. You will need to find someone else who can."
This conversation was one I would have with senior officers throughout the early years of my career.
Army Regulation (AR) 600-100: Army Profession and Leadership Policy defines a mentee as a person of "lesser experience" and a recipient of "advice and counsel." The mentee must value the "assessment, feedback, and guidance" given by the mentor. The regulation defines the mentor/mentee interaction not as a singular episode, but as a "voluntary developmental relationship" characterized by "mutual trust and respect."
Army Doctrine Publication (ADP) 6-22: Army Leadership and the Profession describes the relationship between mentor and mentee. One description alerts the mentee to the most challenging and frequently misunderstood approach to this relationship: "The developing leader often initiates the relationship."
The first mentorship obstacle I encountered as a young lieutenant was one of expectations. I expected a senior officer to ask me to be their mentee. Over the years, many of my peers confessed the same. When a young officer is advised to find a mentor, the immediate assumption is that the one pointing out the need is also volunteering to fulfill the need.
I waited years for a more experienced officer to see my need for coaching and development, then decide to mentor me. After years of waiting, I began to wonder why no one wanted me to be their protege. I misunderstood my role and the responsibility I had to initiate the relationship.
Step 1 to becoming a mentee is to request a mentor. This request does not need to be a formal conversation and does not require the words, "Will you be my mentor?" Initiating mentorship can be as simple as asking for career guidance, asking for pros and cons of a duty station or even asking for advice on tactics. Mentorship starts with a question.
In late 2018 at the National Training Center at Fort Irwin, California, I asked a senior officer, not in my chain of command, a tactical question: "Where do you place yourself on the battlefield, and why?" I was genuinely curious and eager to gain insight from him. His answer, coupled with a history I had with the senior officer, led me to desire a mentorship relationship with him.
We never had a formal conversation where I asked him to be my mentor, and he never asked me to be his mentee. However, I felt the liberty to continue seeking advice, and over the years, we developed a relationship based on mutual trust and respect extending to personal and professional growth.
Develop a Relationship
AR 600-100 and ADP 6-22 define the relationship between a mentor and a mentee. ADP 6-22 says it is a relationship existing outside the chain of command, or "outside the superior-subordinate relationship." The language in this doctrine identifies well the thesis of the second mentee obstacle: Mentorship requires a personal relationship between the mentee and mentor.
A second step to becoming a mentee is to ask the right person, and in turn, to not ask the wrong person. Doctrine defines the parameters as outside the chain of command, because the nature of true mentorship requires the mentor to aid in personal and professional growth. A mentorship relationship of this kind is inappropriate if the senior leader is also the rating or senior rating officer (or NCO) in the mentee's rating chain. The senior leader would not be able to render a fair and impartial objective evaluation when the evaluated soldier is the senior leader's protege. It generates the appearance and substance of bias.
Moreover, the mentee should not seek a mentor in someone with whom they do not, or will not, have a relationship. This includes senior leaders they have just met and with whom they will never spend time. ADP 6-22 describes a frequent arrangement where the "relationship extends past the time where one party has left the other's chain of command."
In this scenario, the junior soldier enters a unit, builds professional relationships with senior leaders, and only once they depart the formal rating chain is it acceptable to initiate a mentorship attempt with a senior leader. The developing leader may also initiate if the senior leader departs the chain of command before the junior leader does.
The mentee's goal in a mentorship relationship is to achieve personal and professional growth—to gain competence. A healthy mentorship relationship is one with mutual trust and respect, and honest two-way communication.
Lastly, ADP 6-22 defines the mentor-mentee relationship as voluntary. Therefore, if both parties do not have mutual interest, the relationship will be of little to no value. It is incumbent upon the senior leader to make an honest assessment of their position, time and ability to invest in a potential protege.
A mature, competent, professional leader will know when to refrain and when it is appropriate to enter a mentorship relationship with a developing leader.
For this reason, a near-retirement lieutenant colonel met me and quickly observed my need for a mentor, then promptly affirmed his inability to serve in the role. He was right to do so.
Avoid Confusion
The third mentee obstacle is a matter of quantity and quality: While one mentor is not enough, too many mentors can lead to confusion. ADP 6-22 does not define the quantity of mentors a mentee should have. This is a matter of prudence and work-life balance. ADP 6-22 does, however, describe four types of mentorship: personal, professional, supportive and spiritual.
The third step to becoming a mentee is to define personal goals, understand strengths and weaknesses, and find mentors who can speak to more than one area of life. Mentees who limit themselves to one mentor misunderstand the human factor; one mentor does not have the time and expertise to mentor well in each of the four types of mentorship.
To accommodate the shortcomings of a single mentor, the Army mentorship concept is based upon relationships between senior and junior leaders, between peers, between senior NCOs and junior officers, between civilian leaders and uniformed personnel, and even between chaplains or spiritual leaders and individuals who desire spiritual mentorship.
The four types of mentorship are of themselves not all-inclusive but do well to shape the total soldier. A mentee should attempt to find mentors who can provide growth and competence, understanding that mentors also have their own strengths and weaknesses. The mentee should tailor their expectations to accommodate their mentor.
Checks and Balances
Sometimes mentees are tempted to accept every piece of mentorship and advice from senior leaders. According to ADP 6-22, "proteges carefully consider assessment, feedback, and guidance." This doctrinal admonition reminds the mentee to weigh and measure guidance in accordance with their own goals and aspirations.
Mentors are not always right. Therefore, having more than one mentor can add healthy layers of checks and balances to the mentorship process.
Army mentorship is organic; it is voluntary. Although there is no doctrinal mandate to serve as a mentee, it is a sign of wisdom to seek and accept mentorship from a more senior leader. To be a mentee is not a sign of weakness or incompetence. Mentees have the unique opportunity to humbly request a mentor, engage in multiple healthy mentorship relationships and gain competence in the profession of arms. A competent Army requires mentees and mentors at every rank, role and echelon.
Capt. Dallas Meachum is an assistant professor of military science, Army ROTC, at the University of Maine. Previously, he commanded a watercraft maintenance company at Fort Eustis, Virginia. He has a master's degree in transportation and logistics management from American Military University.
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Wildlife Images is happy to discount school admission to $5 per student! Visit our school field trip page for information and scheduling.
Available by advance reservation for groups of 15 or more. Call 541-476-0222 to arrange your visit and take advantage of a 10% discount.


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Q: Problem with plugin cordova-sqlite-storage I have a problem with a project of cordova (Ionic), the app compile sucefully. But when I add cordova-sqlite-storage plugin.
cordova plugin add cordova-sqlite-storage --save

the compile break.
└────╼ ionic cordova build android
> ionic-app-scripts build --target cordova --platform android
> cordova build android
9.0.0
cordova-android-support-gradle-release: Android platform: V7+
cordova-android-support-gradle-release: Wrote custom version '27.+' to /home/dylan-roman/Documentos/projects/adnmovil/platforms/android/app/build.gradle
cordova-android-support-gradle-release: Wrote custom version '27.+' to /home/dylan-roman/Documentos/projects/adnmovil/platforms/android/cordova-android-support-gradle-release/movil-cordova-android-support-gradle-release.gradle
cordova-android-support-gradle-release: Android platform: V7+
cordova-android-support-gradle-release: Wrote custom version '27.+' to /home/dylan-roman/Documentos/projects/adnmovil/platforms/android/app/build.gradle
cordova-android-support-gradle-release: Wrote custom version '27.+' to /home/dylan-roman/Documentos/projects/adnmovil/platforms/android/cordova-android-support-gradle-release/movil-cordova-android-support-gradle-release.gradle
Checking Java JDK and Android SDK versions
ANDROID_SDK_ROOT=/home/dylan-roman/Android/Sdk (recommended setting)
ANDROID_HOME=/home/dylan-roman/Android/Sdk (DEPRECATED)
Using Android SDK: /home/dylan-roman/Android/Sdk
Subproject Path: CordovaLib
Subproject Path: app

> Configure project :app
WARNING: Configuration 'compile' is obsolete and has been replaced with 'implementation' and 'api'.
It will be removed in version 5.0 of the Android Gradle plugin.
For more information, see http://d.android.com/r/tools/update-dependency-configurations.html.
+-----------------------------------------------------------------
| cordova-android-support-gradle-release: 27.+
+-----------------------------------------------------------------

> Task :app:stripDebugDebugSymbols FAILED

FAILURE: Build failed with an exception.

* What went wrong:
Execution failed for task ':app:stripDebugDebugSymbols'.
> No version of NDK matched the requested version 21.0.6113669. Versions available locally: 22.0.6917172

* Try:
Run with --stacktrace option to get the stack trace. Run with --info or --debug option to get more log output. Run with --scan to get full insights.

* Get more help at https://help.gradle.org

Deprecated Gradle features were used in this build, making it incompatible with Gradle 7.0.
Use '--warning-mode all' to show the individual deprecation warnings.
See https://docs.gradle.org/6.5/userguide/command_line_interface.html#sec:command_line_warnings

BUILD FAILED in 2s
40 actionable tasks: 11 executed, 29 up-to-date
Command failed with exit code 1: /home/dylan-roman/Documentos/projects/adnmovil/platforms/android/gradlew cdvBuildDebug -b /home/dylan-roman/Documentos/projects/adnmovil/platforms/android/build.gradle
[ERROR] An error occurred while running subprocess cordova.

        cordova build android exited with exit code 1.

        Re-running this command with the --verbose flag may provide more information

If I remove the plugin compile sucefully
I allready try remove the plataform and then add android and cordova prepare
Please help me

A: I had some simliar problems with cordova on Mac OS while compiling for ios. I switched to cordova-plugin-sqlite-2 and it fixed the problem for me. You dont need to made any code changes, just install this plugin instead of the other sql plugin.

redpajama_set_name: RedPajamaStackExchange


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Drinks with films
Celebrating cinema with cocktails!
46th Telluride Film Festival — Racing thru Four Days of Films
Posted on September 5, 2019 by Jill
A gathering of the luminaries at the 2019 Telluride Film Festival
Film Festivals have their own zeitgeist. Cannes has glamour, Sundance has snow and celebrities, and Telluride has a mountain top cathedral for film aficionados. Programmers fight to have films premiere at their festival and attracting celebrities is very important. Film guests sell tickets but also create a buzz at the event. Some festivals like the Mill Valley Film Festival and the Telluride Film Festival have less work to do to attract the big names — both events are in beautiful places and more importantly, attract fans that will allow filmmakers and stars to walk the streets sans bodyguards and publicists.
This year's Telluride Film Festival was graced with the presence of such luminaries as Martin Scorsese, Adam Driver, Philip Kaufman, Bong Joon-ho, Edward Norton, and Renee Zellweger. Long-time Festival favorites, Werner Herzog and Ken Burns brought new works to the Festival and first-time festival attendees like Antonio Banderas (Pain and Glory) professed their love for the magic of Telluride. Programming was strong. There weren't as many thought-provoking or life-affirming features as in years past but there was certainly a breadth of subjects covered. From Imelda Marcos (The Kingmaker) to Oliver Sacks (Oliver Sacks: His Own Life) to sports (cycling, soccer, Australian football, race cars) and hot air balloons (The Aeronauts); from portrait painting (Portrait of a Lady on Fire) to art forgery (Lyrebird).
What was interesting was to hear how many film lovers either LOVED or HATED some of the films. Everyone was excited to see the Hollywood films: Ford v Ferrari, Judy, Marriage Story and Motherless Brooklyn but the more offbeat Uncut Gems? Even with the selling point of having Adam Sandler there for a lively Q & A, Uncut Gems warranted a lot of walk outs. Directors Josh and Bennie Safdie tapped Oneohtrix Point Never who also scored their 2017 film Good Time; both films feature a frenetic soundtrack. Sound and scores were an important part of the the film experience this year and Uncut Gems soundtrack was a cacophany that may have reflected the character's state of mind — but it was difficult to endure.
There were 30 main film programs, three tributes, shorts programs, retrospectives and outdoor screenings. Guest Director Pico Iyer selected five International film highlighting women in film. With this year's focus on sound, there was a tribute to Dolby Laboratories, two silent films and many films about musicians. Ken Burns unveiled his series on Country Music, documentaries included Billie (Billie Holiday), Amazing Grace (Aretha Franklin), Linda Ronstadt: The Sound of My Voice, Tex Mex music (Chulas Fronteras) and The Gift: The Journey of Johnny Cash. Two themes were evident in the 2019 program — music, there were some unusual scores this year — and long screening times. Of the 30 main film programs, 20 were at or over 2 hours.
Waves, 2019 — A24
One of the longer films that was also divisive is Waves. I spoke to people who walked out, a few people who ranked it as a favorite, and others who felt it was trite and the soundtrack annoying. Trey Edward Shultz (It Comes at Night, Krisha) uses a few unusual filmmaking techniques to put the audience in the mindset of the characters. Not everyone was a fan of the spinning Go Pro shots in the car (used repeatedly) or the full-screen color waves to represent emotions. The soundtrack also acts like a character in the film. It's a hard-driving mix of hip hop and rap by Trent Reznor and Atticus Ross and while the characters are listening and singing to music, the soundtrack is playing something different for us. It was jarring at first and like the color blocks on the screen, took some adjusting to.
Taylor Russell plays the daughter in Waves
A tragedy told from two focal points, the acting is strong particularly from the two young actors Kelvin Harrison Jr. and especially Taylor Russell as the young woman who transforms from a background player who's withdrawn, to the focus of the narrative. Waves is getting a lot of critical acclaim and though I felt it had some beautiful moments, a little judicial trimming would've gone a long way to transforming the film.
Working the festival as a Volunteer (love the Sheridan Opera House crew!), there were many films I didn't get the opportunity to see. So I'm thankful for the After Festival screenings and happy that I had to opportunity to catch Parasite and The Two Popes. Thank you Telluride Film Festival. What a gem of a festival!
This entry was posted in Art House Films, Big Budget Action Movies, Documentary Film, Film Festival, Independent Cinema, Uncategorized, women in film and tagged Adam Sadler, Atticus Ross, Ken Burns, Safdie Brothers, Telluride Film Festival, Trent Reznor, Uncut Gems, Waves, Werner Hetzog by Jill. Bookmark the permalink.


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The Last Words of 15 Historical Figures
Alfred Hitchcock, director
Last words: "One never knows the ending. One has to die to know exactly what happens after death, although Catholics have their hopes."
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You are bidding on a Rare 1959 Cadillac Series 6700 Fleetwood 75 9 passenger Limo. 59S032598. We are matching titles and will update info. The car is original and mostly complete. It is black in color. The car has a V8 engine and no air cleaner. AT transmission. Does not run. The car has 29,849 miles on the odometer. The car is relatively clean, but there still is areas of surface rust and some rust in the lower body. Can not look under. The interior is in good condition and is loaded with options and black leather bench in the front. It has grey cloth seats and jump seats in the rear. It has electric windows and locks and AT. clock. This is a good car and would be a great survivor or for restore. This is part of the Brown Estate in SE Wisc. This is a timed On-LINE ONLY AUCTION. ALL SALES ARE AS IS AND ALL SALES ARE FINAL. NO EXCEPTIONS. SERIOUS BIDDERS ONLY. Money Wire or Verified Funds for ALL Purchases. NO CC accepted. Buyer's are responsible for review of information, pictures, and bid accordingly. There are rare vehicles in this collection for restore. There is video on my Facebook Page VanDerBrink Auctions, LLC. You may also call for with questions 605-201-7005. Buyers are responsible for shipping and removal costs and arrangements at load out. Due to the location, safety, and family privacy, no preview available at this time. The vehicles, Most all do not roll, have flat or no tires. They all do not run. These cars are for restore. Winning Bidders will be given the location and times for pickup/arrangements. The family will be happy to tell winning bidders about their Father/Father-in-Law. The location is easily accessible and the cars are easily accessible and load if you have a winch, trailer, and pickup. The location will not let a SEMI transport to drive in. You would need a roll back and then off-load onto the truck on the road. I am having an expert review the cars and will add those comments after his inspections. This is a very fun collection of Rare and hard to find cars and you will be the talk of the town with these barn finds. Happy bidding and rememeber serious bidders, Please.


Christianity is in danger of ceasing to be a truly global faith as increasing numbers of its followers flee violence and persecution across swaths of the Middle East and Africa, according to a new report.
"Christians are fast disappearing from entire regions – most notably a huge chunk of the Middle East but also whole dioceses in Africa. In large part, this migration is the product of an ethnic cleansing motivated by religious hatred," says Persecuted and Forgotten?, published on Tuesday by the Catholic campaign group, Aid to the Church in Need (ACN).
Christianity was "changing from being a global faith to a regional one, with the faithful increasingly absent from ever-widening areas".
Its latest report, covering the past two years, concludes that the difficulties facing Christians have worsened in 15 out of 19 countries under review. Militant Islamism is the main – and increasing – threat, but Christians have also been targeted by extremists of other faiths and totalitarian regimes such as North Korea.
The report repeats claims that Christians are being "driven out of [the church's] ancient biblical heartland" of the Middle East, saying they are "on course to disappear from Iraq possibly within five years – unless emergency help is provided at an international level on a massively increased scale".
In Africa – described as "the one continent which until now has been the church's brightest hope for the future" – the rise of militant Islamic groups in countries such as Nigeria, Kenya and Sudan is destabilising Christian presence, the report says.
It says that Christians are the most persecuted faith group in the world, citing the Frankfurt-based International Society for Human Rights 2012 report, which estimated that 80% of all acts of religious discrimination were against Christians. The Centre for the Study of Global Christianity in the US estimates that 100,000 Christians die every year, although some question the legitimacy of this figure.
The ACN's report says that in the past two years, the number of countries in which Christians suffered extreme persecution was 10, up from six in the period covered by its previous report. Iraq, Nigeria, Sudan and Syria joined China, Eritrea, North Korea, Pakistan, Saudi Arabia and Vietnam.
It said 120,000 Christians had fled Mosul and Ninevah in Iraq after a takeover by Islamic State in the summer of 2014, and by May 2015, 100,000 Catholics had left the Nigerian diocese of Maiduguri in the wake of attacks by Boko Haram. It also warned of rising anti-Christian attacks by militant Buddhists in Burma and Sri Lanka.
The Vatican sent ACN a message of support for the report's launch on Tuesday, saying Pope Francis appreciated efforts to keep "before the world the plight and suffering of Christians persecuted for their faith".
David Cameron, the UK prime minister, also spoke out against systematic discrimination against Christians in a message for the report's launch, saying the government was committed to promoting religious freedom and tolerance in the UK and around the world. "Now is not the time for silence. We must stand together and fight for a world where no one is persecuted because of what they believe," the prime minister said.
ACN has launched fundraising campaigns to assist Christian refugees from the Middle East and elsewhere, although some critics have said all refugees should be eligible for help regardless of their religion.


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Эхопроцессор ( — «обработчик эхопочты») или тоссер (, от глагола ) — компьютерная программа для обработки эхопочты в Фидонете. Тоссеры обычно могут обрабатывать и нетмейл, но эта функция часто отдаётся трекеру.

Функции
 Тоссинг () — распаковка полученных мейлером пакетов, и размещение сообщений из них в базе сообщений (например, в формате ). Сообщения, предназначенные для других систем, могут не записываться в базу, а сразу упаковываться для них — см. ниже. Сообщения, которые уже обрабатывались тоссером (их MSGID сохраняются в специальной базе), могут удаляться либо помещаться в специальную область сообщений.
 Сканирование и упаковка () — поиск в базе новых сообщений, подлежащих отправке, и создание пакетов для пересылки мейлером. Большинство эхопроцессоров в списке кодов продуктов FTSC называются «» («упаковщик»).
 Очистка или упаковка базы () — поиск и удаление сообщений, помеченных как удалённые, и старых сообщений. Часто выносится в отдельную программу.

Известные тоссеры

Примечания

Программное обеспечение Фидонета

redpajama_set_name: RedPajamaWikipedia


It will be ideal here to trace the African history and political development to uncover some of the foundational elements of the major problems confronting Africa. The most conspicuous symptom of Africa's afflictions is its slow development-economically, socially and politically. Moreover, this issue is underscored by Africa's long-enduring platform of compelling concerns: lack of long-term unity and consistent application of agreed policies among member countries and their leaders; high-scale conflicts; environmental problems; corruption and governmental mismanagement. It is my belief that these unresolved matters markedly impede the AU's ability to perform its objectives for a more unified and advanced Africa. This paper explores this thought. The history, cultures, political and economic developments of the member nations that constitute both the AU and the OAS are explored in this chapter as a base for the ideas and observations set forth in the rest of this work. The Chapters 1 will focus on history, culture, politics and economy of countries in both organizations, and comparative analysis of the structures of both organizations. Chapter 2 deals with The Structural Examination of OAS and AU. Chapter 3 will focus on case study of the conflicts in Burundi and Colombia and relevant be dedicated to basic fundamental human rights cases in OAS and AU. Chapter 5 will contain the Conclusion.
Akingbulu, Samuel Bamidele, "A Structural Comparison of African Union [AU] and Organization of American States [OAS] Through the Lens of Conflict Management and Human Rights" (2014). Theses and Dissertations. 57.


Reached Jalandhar cantonment around 6:15 in the morning. Just as were regrouping and deciding our next course of action, we felt some T-rex like vibrations of Jurassic park and high pitched screams from an unmistakably electric loco as the New Delhi bound Amritsar Shatabdi express decimated Jalandhar cantonment with pure aggression and demonstrating the doppler effect to perfection. The closed environment of the railway station made the experience very sinister to say the least. Hired a couple of autos and got ourselves dropped near Jalandhar City station and checked into a budget lodging for freshening up and getting ready for the visit. A few of us decided to visit the station to evaluate and gauge the distance and time it would take to reach the station from our lodging. Having gone all the way to the station, we decided to get the tickets for our train to Kapurthala RCF as well to avoid last minute panic. Quickly returned to our lodge, freshened up and with all documents ready, made our way to the station. Our local passenger is scheduled to leave for RCF at 9AM. Breakfast & tea was quickly followed by a spirited dash in search of our DEMU only to wait for the next 15 minutes for it to depart.


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Religion and Spirituality Discussion
True Story: I saw the Devil in the Desert--A Discussion/Introduction
Thread: True Story: I saw the Devil in the Desert--A Discussion/Introduction
21 Mar 2020 03:08 #1
iamonarch
Without sacrifice, nothing's worth it
--Hello everyone. New member, first thread--just wanted to get some opinions on some things and kinda' introduce what it is that brought me here. I'm not really sure if there will end up being a quesiton in here, but there may also end up being several. All in all, I just want to get some insight from people with the types of perspectives that generally characterize this forum's userbase X D
--In 2004/05, the internet really started to take off. I was raised in a Christian household and grew up in a small town, so I didn't really have any external influences except what I saw in movies, video games, etc., and we all know the views expressed in the mainstream are very generalized and somewhat sterile. I still remember the first time I ever went on Wikipedia. I was looking up "the Divine Comdedy," which is an epic poem by the legendary Italian poet, Dante (he has a last name, but like Beyonce and Slash it's hardly necessary). I actually didn't know what to make of that poem--I was so young and so...uninformed or underexposed (like everyone before the internet) that I actually thought it was a true story in some strange way, probably because the majority of my exposure to Christianity had been the book of Revelation (it has dragons and crazy things going on, so of course it caught the attention of a 15 year old, especially when he'd been told it was all "real"). The entire thing just blew my mind, and I couldn't wrap my head around how all of this was supposed to be taken seriously...So, I decided that instead of just taking it all at face value, I'd test it out. One of the things my mom had always brought up for some reason (probably because we all watched Buffy the Vampire Slayer a lot) was that the Book of Revelation said witches, warlocks, and anyone who practiced magic would be cast into Hell without question as such things were an abomination before the Lord. I figured that must mean such things as magic were actually real in some way, because why would God condemn make believe fairy tale nonsense? Therefore...what better way to see if this was all really real then, well, magic.
--I apologize if this is way, way too long already...I just...I wanted to be as clear as possible. This is all the truest thing I could possibly tell anyone, and it's been the single biggest factor in my life since...well, since it happened 14 years ago. Again...maybe I just wanna' know what others think, or maybe if they'd experienced similar things, or maybe...Well, I don't know, but here I am, and I hope you're all still with me.
--So then I...between the years of 16 and 18...got really, really into the occult (I still am, just in a much different way). It was all basic introductory, The Craft level "dabbling," though--nothing serious at first. Candles, sigils, seances, braiding hair, burning incantations...so on and so forth. It really didn't get too intense until we got a Quija board, and I know...it's a board game made by the people that make Monopoly, but we sure as hell treated it like...well, like most people treat it (I think that's what makes it more than just a board game).
--We started using the Quija board, like, every day. We would consult it before doing even mundane things like going to the store or out to eat. We'd break it out at parties, and in our spare time (which was all the time because we were in high school) we'd take it to graveyard and talk for hours; there was this big black tomb stone with a single name in huge, white letters--BEKO--that we sat beside every time (we did this so often we could find our way there in the dark without any issues whatsoever).
--A lot of weird things happened...but I just shrugged it all off because I didn't actually see any of it. One time, for instance, I was sleeping with my girlfriend...She was 16 going on 17 and I'd just turned 18, so I'd moved out into a friend's apartment and was staying in their extra bedroom. I was falling asleep, but she was wide eyed with the blankets up to her chin, whispering to me in a panic: "El, there's something in the room."
--"What?"
--"A shadow--I don't know, but it's just staring at me."
--"Where is it?"
--"At the foot of the bed."
--"Oh...as long as it's at the foot of the bed, it's fine," I'd told her. "It's only a problem if it's beside the bed."
--I don't know where I'd heard that, but that just goes to show how...uncaring I was about the whole thing--unaffected, really. I knew what I wanted, and it wasn't shadows or weird noises or power bumps or anything like that--I wanted to know if the devil was real, and by extension if God was real...then I could know--truly know--that it was a good idea to be a Christian. I just wanted to know.
--Another strange thing, just real quick, was when I'd decided to move out of that apartment. We'd packed everything up and, again, my girlfriend and I were lying in bed, talking about what was gonna' happen next. This odd tapping sound kept happening as we were talking, but we ignored it until finally she was like, "What the hell is that sound?" Getting up to investigate, and I swear this is true, we realized it was the Quija board's little planchette thing knocking against the inside of the box (my girlfriend had made a special board for us in shop class out of high quality wood with a fancy box, almost like a wooden briefcase). We were...mildly amused by this...and just kept talking, eventually falling asleep.
--Then...well...after we'd gotten a little more serious about the magic we were doing--we'd moved into "ceremonial magic," and the "highlight" of that was my girlfriend and I breaking into this old abandoned mansion on the hill overlooking the town (K.C. Hall in Tonopah, NV--I'm totally serious) and spending, like, eight hours straight drawing this incredibly detailed and complicated sigil for an evocation taken from the Lesser Key of Solomon (I don't know if this is related, but every attempt that's been made to renovate and reopen/resale that place has failed miserably, losing people millions in some cases)...Yeah, after that...
--I had a friend named Rena...and her boyfriend, who she'd met online, came down from Reno to see her--we called him "Twitch." He came over to our apartment, and without even knowing the guy at all he came out from the bathroom and immediately said, "Dude, your house is haunted. I was washing my hands and saw this black shadow over my shoulder--like, really." Like before, though, I didn't believe him--I thought his girlfriend, Rena, had told him about what we were in to and he was messing with us. After all, I'd never seen anything...yet.
--So...we all decided to go out of town...and we had a good time...and on the way back from California, driving through the Sierra Nevada mountains, Rena decided to break up with Twitch in the back seat of the car. Clearing the mountain pass, we pulled off the road to let him get some air and he just...took off in to the desert. We waited...and waited...and the sun started going down...so me and another friend (Ashley) decided to go look for him. Now I've thought a lot about why it was Ashely who saw this thing with me and not, say, my girlfriend, but I think it's because, like me, Ashley was raised Christian. Ashley really believed. Also...I guess at a party I'd threatened Ashley with a...uh...like, metal saw or big ass nail or something when her and I had gone into the shed out behind the homeowner's house to get more alcohol (she said that I said the tool--whatever it was--was a present from "them" because she didn't believe and kept making jokes about all of it, but I...don't remember any of that at all...I actually had a lot of things like this happen, as for instance that night the planchette kept knocking against the box I later woke up standing over my the bed of my girlfriend's sister...and I don't remember getting up or anything at all that time either...but again...I just ignored these things...or really...it was more like...well...well...I could, like...hear things, I guess...like...I'd have thoughts and I knew it wasn't really me thinking it, but, well, "them," whoever they were...the spirits, I guess...and I'd be...compelled to listen, I guess...like I wasn't really able to control what I was doing, but I knew I was doing it because they wanted me to--little things, which maybe aren't so little, like not eating properly or not...just...I don't know--I don't really think about that stuff...It's like I can't really think about it anymore...like my mind just won't go there, or...ANYWAY).
--Okay, so...Ashley and I--
--(This is a rambling mess and I'll be incredibly surprised if anyone makes it through this X D)
-- --went to look for Twitch in the desert. It was twilight...still bright enough in the sky, but the ground was becoming an array of shadows...And some odd things happened: We passed this old foundation where a house once stood (in the middle of the desert off a road in the middle of nowhere). It was essentially a rectangular depression encased with a concrete wall about three feet high, ya' know? Full of snakes...like, all kinds of snakes, but not like crawling all over each other, just...here and there in the corners, some dead, but others not so much--I almost stepped on one, and looking around in a panic was when we saw the foundation...It was strange, for sure...but we kept going (I'm beginning to see how incredibly stupid or just incredibly determined we were to not see how absolutely insane things were getting...like WTF, right? X D). Then we kept hearing this really weird noise, like a boar, or a roar of some kind--three times we heard it while walking through the sage brush over little hills, but we kept going cause, well, we had to find Twitch and get home, didn't we?
--Then, we came to this little cliff and climbed down the rocks, and out in front of us was a hill that was bigger then the rest. About halfway up the hill was a group of rocks, and on the rocks was what we thought was Twitch. He'd been wearing black--black pants, black hoodie--and it looked like he was sitting with the arms around his knees. He was far up the hill, and it was dusk so the light was sketchy, but yeah, it was him sitting there--I could see his arms around his knees. So we called out to him. We walked closer and called out again. He stood up...
--(Every time I try to tell people this I get this terrible feeling like all manner of crazy things are just gonna' pop up on me at any second--it's terrifying)
--He started...swinging at the air--like he was furious, throwing a fit, a tantrum even. Flailing the way people flail in mosh pits. Ashley and I were just like, "Wow...okay...understandable I guess." And the thing is, we could see this--we could see that he was standing there throwing his arms about on this f***ing rock.
--(This is the weirdest s*** ever, guys, just thinking about it makes my eyes water)
--And then...he just...He kept swinging, but it...slowed down...like...graceful--like dancing. Keep in mind, he was wearing all black, hood up, and it was sunset and there were shadows and he was far away up this hill, but still...we could see him. And then...he started to, like...tumble...like...cart-wheels...but...as he went over onto his hands, his hands seemed to become his feet, because even though he went over he'd just start swinging again...like he hadn't...like his hands had become his feet and vice versa...twisting...SWAYING, is perhaps the best word...Dancing...but IT WAS TERRIFYING.
--Ashley and I just stood there, absolutely dumbfounded, not saying a word, not moving an inch. Even while I was watching it, all I kept thinking was, "That's not possible. That's not real. Things can't do stuff like that," but not really freaking out...like a joke or a prank or a magic trick when you just think, "What...that can't happen." But there wasn't any magicians in the desert--this wasn't a d*** Criss Angel show in Vegas...and so we just stood and watched...for a long time, though I don't know exactly how long...and then I asked her, "Are you seeing this?"
--"Yeah--WTF is he doing?"
--"I don't know," was my reply, and you see we still thought that somehow it had to be Twitch...after all, what else would it be? So I stepped forward, and it stopped.
--Mid motion, poised to sway, it froze and looked right at us. We were already in awe, so we didn't really react, just remained still and silent, and then the d*** thing on the rock crouched down. Slowly, it started to crawl off the rocks down towards us, and as it did it seemed to change shape...growing in some ways, getting smaller in others, until at last we were sure that what we were looking at was a bear. It WAS a bear, climbing down the rocks, coming to get us, and at that moment we started to seriously lose our s***.
--"Is that a f***ing bear!?"
--"IDK, do they have bears in the desert??"
--"WTF, man, what do we do!?"
--"I don't know--run??"
--Aha...yeah, but we still didn't move at all. It looked so much like a bear, far off up the hill on the rocks, that we'd completely forgot about the fact that only moments ago it was flailing around on thin legs like a human. Now, it was a bear, and maybe because we could actually rationalize that--a bear--our minds latched on to that...
--But it wasn't a bear...because when I dared to take back the small step I'd made earlier...it stopped. Dead in its tracks, it started to move backwards back up the rocks, STOOD UP on its HUMAN legs, and then sat down, putting its arms around its knees...just like when we'd first came upon it.
--We hurried back up the small cliff face, and RAN BACK TO THE CAR. About half way back we ran into my girlfriend and the others (Rena and Jennie). Panic stricken, we told them what we'd seen, and Jennie, the fool, was like, "I want to see it," but we screamed, "NO! We're going back to the car, NOW!"
--When we got back, we realized that Twitch had walked out a little ways, turned in a wide arc, and came right back to the car. He'd been there the whole time.
--...I'm not asking what that was (though any opinions are more than welcome). I'm not asking why it appeared out there and not, say, in the magic triangle in the abandoned mansion (though thoughts on that would be nice, as well). All I'm really doing...is saying that I saw something that I KNOW isn't..."part of this world." There's nothing--NOTHING--that I know of that can behave in the way in which that thing behaved...So then I know now...that things like that exist...but I don't know for certain what it was...where it came from or why...or what it all means...and I've spent the last 14 years trying desperately to figure that out...
--I have...no social life...I work...I've had girlfriends and relationships...I'm an artist...I have hobbies...but more and more everything and everyone seemed to just...fall away...because, in part, of my obsessive quest to figure out what the hell that thing was, what it means, and why we saw it out there...I've spent eight hours a day for the last seven years or more researching every book and video and blog I can on religion, spirituality, mysticism, history, science...philosophy, anthropology, psychology, theology and various ideologies...on and on and on...
--Because...even thought I saw it...it doesn't prove any one idea over any other idea. Say it was a "demon." Okay, but does that necessarily prove the bible? No, because what if it was a skin walker? Does that prove Native American beliefs? A vengeful ghost? Does it prove how and why such things should be a reality at all? The Big Bang? God? The Devil...? It doesn't prove anything except that there are things out there...that are not part of our every day experience...and for which we have little to no comprehensive data...
--Confusion...and so I fully understand Crowley's last words: "I am perplexed..."
--...But...I know, like, a ridiculous amount of stuff on all of that stuff now...so...I'm looking forward to being a part of this community.
--Thank you to anyone who made it through all that. I'd love to hear what you all have to say.
Gabriel9
Roman Catholic Christian
Re: True Story: I saw the Devil in the Desert--A Discussion/Introduction
This is normal...
The other day I called the demons of a girl I know, to see if there was something powerful in her, and I was like... opening the window full, and doing the summons, and nothing would happen, and my dog was there beside me, they I said, come through earth, sky, and ALL DIMENSIONS' they they came VLAM my dog was completely startled and jumped, out, I didn't feel them yet, then I was like,, it failed, and I called her, she was depressed, wanted to get rid of everyone on earth, just like my minor demon was in her, I was like ok, it failed.
But then 2 hours later, haha they were there, making me feel awesome!!! best time by far since years, I let them inside me all night and nothing really happened, I have protection devices around my bed, and in the morning I just cast them out, she was then back very happy lol,
I saw God also, so there is no problem seeing what you saw, for sure we do not know what they really are, maybe UFO's
What is for sure is that when someone channel my demons fucking weird stuff happens, in the later years. I am not possessed by definitely extremely connected and it is not my desire, like yours,
If I had a desire to conjure devils demons etc, I would be totally freaked out.... I so far only summon 'succubus' 2 times and once they just appeared like that... 100% invisible to others.
However I can tell you that since 20+years some evil murderer spirit escaped from hell controls and feeds upon all witchcraft and ouija boards, I would never use any knowing how strong this spirit must be by now...
Do small things like call human demons in yourself, but do not play with spirit summoning, I feel that at this our the demon themselves are being destroyed by that spirit...
I just cannot believe that even hell had no power on this thing years ago, it killed many 'real' witches since then... true.
Btw you could have communicated with this 'thing' if you believed in God, you would feel how powerful it is and how far it wants to go to bring your soul in hell by any means
It might come back in your life, just do not focus or think about it. Once you encounter it you have the signature in you. The closer to LOVE, the more pure your heart is the more these things will seek you naturally.
Last edited by Gabriel9; 21 Mar 2020 at 17:11.
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