text:
\section{Introduction}

Experiments exploring quantum physics have reached impressive levels in the control and the engineering of systems with increasing complexity and coherent degrees of freedom ~\cite{AruteNATURE2019, ZhongSCIENCE2020, EbadiNATURE2021}.
The development of techniques for the characterisation of quantum states, their dynamics and possible measurement observables is motivated by ground-breaking applications in communication, computing, sensing and simulation.
Yet, the exponential complexity of quantum systems poses significant challenges for their efficient description.
This quest is currently stimulating a broad and interdisciplinary community to develop novel learning algorithms and techniques, with various levels of approximation. Learning quantum systems in a practical way is crucial for harnessing efficient techniques for quantum control and optimisation, broadly applicable to a variety of physical platforms such as spin qubit, trapped ions, photonic and atomic systems, as well as superconducting circuits.
For instance, in the context of quantum computation, it is important to calibrate and benchmark quantum gates and operations implemented in the actual device, and to prepare qubit states with high fidelity.
In quantum sensing, while learning quantum states and their dynamics is not strictly necessary, it is nevertheless crucial for the optimization of quantum state preparation, parameter encoding and measurement, with the target of improving  the accuracy and precision of the device.
Furthermore, tomographic methods can facilitate efforts in quantum simulations, to understand properties of ground and excited states in the systems and/or to categorise quantum phases of matter.

While the full simulation of large quantum systems may only be possible with a quantum computer\cite{Feynman1982}, classical methods for the characterisation of systems of medium size \cite{Preskill2018} (e.g. of the order of hundreds of qubits) are currently playing a key role.
Such an approach is currently applied with great success in experiments, also shedding new light on fundamental properties such as entanglement, quantum superpositions and nonlocality.

\begin{figure*}[ht]
\centering
\includegraphics[width=.95\linewidth]{Fig1.pdf}
\caption{\textbf{Learning quantum states, dynamics and measurements.}
\textbf{(a)} In this review, we divide the task of learning quantum systems into the sub-tasks of learning quantum states (Sec.~\ref{sec_states}),  quantum dynamics (Sec.~\ref{sec_dynamics}),  quantum measurements (Sec.~\ref{sec_measurements}), and optimisation techniques  (Sec.~\ref{sec_optimisation}).
\textbf{(b)} A list of widely-used methods for learning quantum states (top), dynamics (centre), and measurements (bottom), indicating the number of qubits $N$ each method has been applied to for experimental and simulated data. The compared methods are (Maximum-likelihood~\cite{haffner_scalable_2005}; MaxLik), (compressed sensing~\cite{riofrio_experimental_2017}; CS), (permutationally-invariant QST~\cite{schwemmer_experimental_2014}; PI-QST), (tensor-network QST~\cite{cramer_efficient_2010,lanyon_efficient_2017}; TN-QST), (restricted Boltzmann machines~\cite{torlai_neural-network_2018,torlai_integrating_2019}; RBM), and (classical shadows~\cite{elben_mixed-state_2020,huang_predicting_2020}; Shadow) for quantum states, (standard QPT~\cite{Altepeter_2003,Rodionov_2014,torlai_quantum_2020}; SQPT), (random benchmarking~\cite{mckay_three_2019}; RB), (compressed sensing~\cite{Shabani_2011,Rodionov_2014}; CS), (tensor-network QPT~\cite{torlai_quantum_2020}; TN-QPT), (quantum Hamiltonian learning~\cite{hou_2019_11qubit,wiebe_bootstrapping_2015}; QHL), (quantum model learning agent~\cite{gentile_learning_2021}; QMLA), (eigenstate-driven hamiltonian learning~\cite{zhu2019reconstructing}; EdHL), (Lindblad tomography~\cite{samach2022lindblad}; LT), and (process tensor tomography~\cite{white2022-PTT}; PTT) for quantum dynamics, and (quantum detector tomography~\cite{ChenPRA2019}; QDT), (joint quantum-stateand measurement tomography~\cite{ZhangPRL2020,KeithPRA2018}; JQSMT), and (gradient descend methods~\cite{hetzel_tomography_2022}; GD) for quantum measurements.
}
\label{fig:fig1_summary}
\end{figure*}

Here, we provide a broad overview of a recently developed suite of different classical learning techniques that provide full or partial knowledge about the properties of quantum systems, including their interaction with the environment, as well as learning ways to better measure and control them.
This review is structured in three main parts, related to learning quantum states, quantum dynamics and quantum measurements (see Fig.~\ref{fig:fig1_summary} for a schematic overview), including a
final section devoted to applications of learning techniques for the optimisation of quantum experiments.
We also provide a succinct background of the key concepts recurring across many of these approaches - such as the Bayesian formalism and \glspl{nn} - and  outline important open questions.
Our overview of the literature targets a highly multidisciplinary audience, including quantum information physicists, statisticians, engineers and computer scientists, which characterizes the community involved in this field. While not covering technical details on the different learning methods, we make sure to provide pertinent references to the reader.

We do not discuss learning approaches based on quantum algorithms, including quantum machine learning, and we refer the reader to reviews on that subject \cite{Schuld2015, BiamonteNATURE2017, dunjko_machine_2018}. Beyond that, there are many existing excellent reviews that cover and develop some other aspects related to our discussion, such as the use of machine learning techniques for quantum systems and for physics more generally \cite{dunjko_machine_2018,carleo_machine_2019,marquardt2021machine,dawid2022modern},  for the foundations of quantum theory \cite{Bharti_2020} and computer-inspired design of quantum experiments \cite{krenn_computer-inspired_2020}.

\section{Learning Quantum States}\label{sec_states}

The reconstruction of quantum states from experimental measurements is crucial for characterising the performance of near-term quantum hardware in terms of fidelity with target states, expectation values of local or nonlocal observables, correlation functions, and other properties.
Accurately learning quantum states is also a key requirement for studying fundamental physics, such as the entanglement of the generated  quantum states or the identification of quantum phases of matter.
In the following, we first discuss the reconstruction of quantum states in a general setting, and then review a variety of assumptions that renderthis task more efficient.
Finally, we discuss methods that address cases for which only particular properties of the quantum states are required, and conclude with an overview on optimal qubit readout.

\subsection{Quantum state tomography}\label{section:QST}

The process of inferring an unknown quantum state $\rho$ from the measurement of multiple identically-prepared copies is known as \gls{qst}~\cite{vogel_determination_1989, raymer_complex_1994, leonhardt_quantum-state_1995,leibfried_experimental_1996,hradil_quantum-state_1997, james_measurement_2001, banaszek_focus_2013, paris_2004}.
Here, we will focus on \gls{qst} of finite-dimensional discrete-variable systems, such as multi-qubit systems.
For tomographic methods in continuous-variable systems, including Wigner tomography to reconstruct the state of an optical mode, we refer to Refs.~\cite{paris_2004,lvovsky_continuous-variable_2009}.

The most-widely used \gls{qst} technique is based on { \bf maximum-likelihood estimation}. This method searches for the quantum state featuring the largest probability, or likelihood, of the observed measurement results~\cite{hradil_quantum-state_1997, banaszek_maximum-likelihood_1999, paris_2004, lvovsky_continuous-variable_2009}.
Here, the likelihood of a series of $m$ measurement results is given by
\begin{equation}
    \mathcal{L}[\rho] = \prod_{j=1}^m {\rm Tr}\left[\rho E_{\mu_j}\right],
\end{equation}
where $E_{\mu_j}$ is one of the $K$ elements of a \gls{povm} (namely satisfying $E_\mu\geq 0$ and $\sum_{\mu=1}^K E_\mu = 1$) corresponding to the $j$th measurement result $\mu_j$.
The maximum-likelihood state, defined as $\rho=\arg\max_{\sigma}\mathcal{L}[\sigma]$, can be obtained by iterative algorithms, such as\cite{smolin_efficient_2012, rehacek2007diluted}
\begin{equation}
    \rho_{i+1}=\mathcal{N}_i \,R(\rho_i)\, \rho_i \, R(\rho_i),
\end{equation}
where $R(\rho)=\sum_{j=1}^m E_{\mu_j}/{\rm Tr}\left[\rho E_{\mu_j}\right]$, and $\mathcal{N}_i$ is a normalisation constant.
Alternatively, one can use {\bf Bayesian approaches} (see Fig \ref{fig:fig2_bayesian}a, Box \ref{sec:bayesian_box}), providing a natural way to include prior information and to compute error bars~\cite{blume-kohout_optimal_2010,granade_practical_2016,granade_practical_2017,granade_qinfer_2017} and confidence regions~\cite{christandl_reliable_2012,faist_practical_2016} for the estimates (see Box~\ref{sec:bayesian_box}).
In real-time \gls{qst}~\cite{mahler_adaptive_2013,qi_adaptive_2017,granade_practical_2017}, the observed data are analysed simultaneously to their recording, and thus {\bf adaptive techniques} such as  self-guided \gls{qst}~\cite{ferrie_self-guided_2014,rambach_robust_2021} can be used to optimise the data acquisition procedure.

\begin{figure*}[ht]
\centering
\includegraphics[width=.95\linewidth]{Fig2.pdf}
\caption{ \textbf{Bayesian inference for parameter estimation and model learning.} \textbf{(a)} Bayesian inference enables to learn the probability distribution for unknown parameters by applying Bayes rules with the measurement outcome of each experiment performed on the quantum system (Box \ref{sec:bayesian_box}). \textbf{(b)} An overview of the principal subtending protocols for learning quantum dynamics that employ a (quantum) simulator~\cite{granade_robust_2012, wiebe_quantum_2014, wiebe_hamiltonian_2014, wiebe_bootstrapping_2015, wang_experimental_2017, hou_2019_11qubit, hincks_statistical_2018, gentile_learning_2021, flynn_2021_qmla}.
For each iteration $i$, identical fiducial states $\ket{\psi}_{\text{trial}}$ are prepared and fed to both the system to be characterised and the simulator, where $\ket{\psi}_{\text{trial}}$ undergoes evolution for a predetermined time $t$. The latter is assumed to be tunable, to dial the trial Hamiltonian $H_\text{trial} (\bm{x})$, whose parametrised terms are symbolically rendered by coloured squares, which is either known (e.g. in \gls{qhl}) or hypothesised (e.g. in \gls{qmla}).
The outcome $\mu_i$ collected at the end of each epoch is then fed into the protocol to perform the relevant inference (see also c).
\textbf{(c)} An outline of the protocols that have been demonstrated for model learning, \gls{qmla} ~\cite{gentile_learning_2021, flynn_2021_qmla}. The information gathered from experiments is stored in a tree structure, whose leaves are candidate Hamiltonians $H_{\text{trial}} (\bm{x})$. The paramatrisation for those Hamiltonians in the active layer is learned by a method of choice. Trained, active candidate models $(H_{\text{trial}, i}, H_{\text{trial}, j})$ are then pairwise compared according to a metric $\mathcal{B}_{ij}$, capturing the relative performance at reproducing the unknown system and stored as edges of the graph. According to the global outcomes, single nodes or entire layers can be discarded, and relevant terms spawned to new candidate Hamiltonians.}
\label{fig:fig2_bayesian}
\end{figure*}

Generally, \gls{qst} should search through the whole set of degrees of freedom of a quantum state, which scale exponentially with the number of qubits $N$.
For larger $N$, this scaling results in inefficiencies of \gls{qst} due to (i) the requirement of increasing amounts of measurement data~\cite{haah_sample-optimal_2017}, and (ii) the increasing complexity of classical post-processing ~\cite{gross_quantum_2010}.
For instance, any tomographic method that can estimate an arbitrary state $\rho$ up to an error $\epsilon$ in trace distance is shown to require $\mathcal{O}(4^N/\epsilon^2)$ measurements settings~\cite{haah_sample-optimal_2017}. These obstacles highlight the inefficiency of general \gls{qst}, already for the current generation of controllable quantum systems~\cite{haffner_scalable_2005}, and the need for new approaches to make \gls{qst} practical.

\subsection{Efficient quantum state tomography}\label{sec:Efficient_QST}

Several approaches have been proposed to overcome the severe scaling of full \gls{qst}.
The techniques we discuss below show an improved scalability whenever the quantum state can be described by a specific ansatz, and enable \gls{qst} of highly entangled quantum states of up to about a hundred of qubits, which is unfeasible for standard \gls{qst} methods.

{\bf Compressed sensing}~\cite{gross_quantum_2010,flammia_quantum_2012} can be applied to reconstruct a state of low-rank, i.e. whose density operator is described by a sparse matrix. Here, random Pauli measurements are used to reconstruct the unknown state via a convex optimisation algorithm. This method drastically reduces the number of measurements which, however, still scales exponentially with the number of qubits $N$.
Compressed sensing can also be used to construct a low-rank estimate that approximates an unknown general state~\cite{riofrio_experimental_2017}

If the state is symmetric under permutations of the qubits, the number of tomographic measurements can be reduced to scale only quadratically with $N$ by employing {\bf permutationally-invariant \gls{qst}}~\cite{toth_permutationally_2010, moroder_permutationally_2012}.
This method can also be used to estimate the permutationally-invariant part of a general state.
Similarly, for systems of identical particles, it is possible to exploit the symmetry of the state to significantly reduce the number of measurement settings
\cite{klose2001measuring,hofmann2004quantum,banchi2018multiphoton,titchener2018scalable}.
Examples include the reconstruction of a high-spin state\cite{klose2001measuring}, or the reconstruction of the state of many bosons in multiple modes\cite{banchi2018multiphoton}.
Moreover, the number of measurement settings can be reduced to one via the use of a polynomial number of ancillary systems
\cite{d2002universal,banchi2018multiphoton,titchener2018scalable,allahverdyan2004determining}.

A third method is the use of {\bf tensor networks} to represent quantum states~\cite{perez-garcia_matrix_2007,cramer_efficient_2010, baumgratz_scalable_2013, lanyon_efficient_2017, wang_scalable_2020}.
Any pure quantum state can be written as a matrix product state
\begin{equation}
    \ket\psi = \sum_{x_1,\ldots,x_N}{\rm Tr}\left[A_{x_1}^{[1]}A_{x_2}^{[2]}\dots A_{x_N}^{[N]}\right]\ket{x_1x_2\dots x_N},
		\label{eq:mps}
\end{equation}
where $x_n\in\{0,1\}$ labels the computational basis of the $n$th qubit, and $A_{x_n}^{[n]}$ are complex matrices of sufficient dimension~\cite{perez-garcia_matrix_2007}.
The maximal dimension of the $A_x^{[n]}$ is called bond dimension, and if it is small, the description of the state and its tomography are efficient, i.e. linear in $N$~\cite{cramer_efficient_2010}.
Even though matrix product states of low bond dimension cannot describe an arbitrary quantum state, they succeed to reconstruct the ground states of many common Hamiltonians.

Recently, \glspl{nn} have been used to perform \gls{qst}~\cite{torlai_neural-network_2018,carrasquilla_reconstructing_2019,morawetz_u1_2020,palmieri_experimental_2020}.
In particular, a wide class of states can be modeled by a so-called {\bf \gls{rbm}} (Box~\ref{sec_NNbox} and Fig.~\ref{fig:NNs}(b)), suitably adapted to the quantum setting~\cite{torlai_neural-network_2018, torlai_integrating_2019, melkani_eigenstate_2020}.
In this description, the parameters of the state are encoded into the weights of a \gls{nn} that can be learned efficiently\cite{torlai_neural-network_2018}.
The ansatz is given by $\ket\psi = \sum_{\mathbf{x}} \psi(\mathbf{x}) \ket{\mathbf{x}}$ where $\mathbf{x}$ labels the computational basis vectors and, up to a normalisation factor,
$\psi(\mathbf{x}) \propto  \sqrt{p_\mathrm{a}(\mathbf{x})}e^{i\log [p_\phi(\mathbf{x})]/2}$, where
\begin{equation}
	p_\mathrm{k}(\mathbf{x})=\sum_{\mathbf{h}}e^{\sum_{ij}W^k_{ij}h_ix_j+\sum_{j}b^k_{j}x_j+\sum_{i}c^k_{i}h_i}
\end{equation}
corresponds to a Gibbs distribution of a \gls{nn} consisting of a visible layer ($\mathbf{x}$) and a hidden layer ($\mathbf{h}$), with weights $\{W^k_{ij},b^k_{j},c^k_{i}\}$ to encode the amplitudes ($k=a$) and the phases ($k=\phi$) of $\ket \psi$, and $\mathbf{h}=(h_1,\dots,h_w)$ where $w$ is the number of neurons in the hidden layer.
\gls{qst} can also be interpreted as a generative adversarial
game between two players (Box~\ref{sec_NNbox} and Fig.~\ref{fig:NNs}(c)), a generator trying to produce convincing approximations to the state and a discriminator trying to distinguish real states from generated ones
\cite{lloyd2018quantum}. As such, classical and quantum \glspl{gan} have been employed for
\gls{qst}~\cite{ahmed_quantum_2020,braccia2021quantum}, with fewer measurements required when prior knowledge is available.

\begin{figure*}[ht]
\centering
\includegraphics[width=.95\linewidth]{Fig3.pdf}
\caption{\textbf{Different examples of neural networks for learning quantum systems.} \glspl{nn} have been employed after the acquisition of measurement data (i.e., in the post-processing) as (a) feedforward \glspl{nn} for the classification~\cite{carrasquilla_machine_2017,gray_machine-learning_2018,gebhart2021identifying} and regression \cite{goodfellow2016deep} of measurement data, and the prediction of quantum dynamics~\cite{flurin2020using}, (b) as \glspl{rbm} for \gls{qst}~\cite{torlai_neural-network_2018,carrasquilla_reconstructing_2019}, (c) as \glspl{gan} for the tomography of quantum states~\cite{ahmed_quantum_2020} and processes~\cite{braccia2021quantum}, and (d) as \glspl{node} for optimal quantum control\cite{schafer2020differentiable,schafer2021control}. Alternatively to NODE, PINNs (that include the differential equation in the cost function) have been used for creating robust quantum gates~\cite{gungordu2020pinngates}.
}
\label{fig:NNs}
\end{figure*}

\subsection{Extracting specific features of a quantum state}

In cases when one is only interested in some specific properties of $\rho$, the intractability of full \gls{qst} can be avoided by tailoring the measurement and post-processing to the specific property.
Most prominently, several methods have been developed to efficiently measure entanglement properties~\cite{guhne_entanglement_2009, PezzeRMP2018} such as entanglement entropies (e.g. using random measurements~\cite{brydges_probing_2019,elben_mixed-state_2020}), metrologically-useful entanglement (e.g  Fisher information~\cite{PezzePRL2009, strobel_fisher_2014} or spin-squeezing parameters~\cite{PezzeRMP2018}),  entanglement witnesses (i.e. measuring the fidelity to an entangled target state~\cite{lu_experimental_2007}), and entanglement measures (e.g. the concurrence~\cite{walborn_experimental_2006}).
Recently, \glspl{nn} have been employed to measure the entanglement between two arbitrary subsystems~\cite{gray_machine-learning_2018} and to identify single-mode nonclassicality~\cite{gebhart2021identifying} without requiring full \gls{qst} (Fig.~\ref{fig:NNs}(a)).
\glspl{nn} also offer promising routes for the identification of quantum phases of matter and quantum phase transitions~\cite{carrasquilla_machine_2017, van_nieuwenburg_learning_2017, deng_machine_2017, rem_identifying_2019, rodriguez-nieva_identifying_2019, carrasquilla_machine_2020, CarrasquillaPRXQuantum2021}, especially in disordered or topological phases, where no conventional order parameter exists.

If the measurements of the unknown state are randomly drawn from a fixed (but possibly unknown) distribution, the outcome probabilities of future measurements can be approximately learned (probably-approximately-correct, or PAC-learned) after merely a linear number of measurements~\cite{aaronson_the_2007,rocchetto_experimental_2019}. This insight has been strengthened in the recently introduced framework of {\bf classical shadows}~\cite{huang_predicting_2020,aaronson_shadow_2020} that can be used to efficiently estimate $\mathcal{O}(e^m)$ target functions ${\rm Tr}[\rho O_i]$ from $m$ measurements of $\rho$. Here, a series of random projective measurements from a fixed ensemble $\mathcal{U}$ is performed.
These measurements average to the quantum channel $\mathcal{M}[\rho]=\left|\mathcal{U}\right|^{-1}\sum_{U\in\mathcal{U}}\sum_{\mu_U}P_{\mu_U}{\rm Tr}\left[\rho P_{\mu_U}\right]$, where $U$ represents a measurement setting (namely, a unitary transformation applied before a measurement in the computational basis) with possible measurement outcomes $\mu_U$, where each outcome $\mu_U$ corresponds to the projector $P_{\mu_U}$ and is observed with probability ${\rm Tr}\left[\rho P_{\mu_U}\right]$. For instance, an experimentally accessible measurement ensemble $\mathcal{U}$ consists of tensor products of single-qubit Pauli measurements~\cite{huang_predicting_2020}. The classical shadow of $\rho$ for the $j$th measurement is then defined as
\begin{equation}
    \rho_s^{(j)} = \mathcal{M}^{-1}[P_{\mu_j}],
\end{equation}
where $P_{\mu_j}$ is the projector of the $j$th measurement outcome $\mu_j$.
Note that since the map $\mathcal{M}^{-1}$ is not a quantum channel, $\rho_s^{(j)}$ is not a valid quantum state.
However, each classical shadow results in an estimate ${\rm Tr}[\rho_s^{(j)} O_i]$ of the $i$th target function.
After a statistical inference step (median-of-means), one obtains the final estimates of the target functions.
In this way, the estimation of $\mathcal{O}(e^m)$ target functions ${\rm Tr}[\rho O_i]$ requires only $m$ random measurements~\cite{huang_predicting_2020}, where the proportionality factors depend on the choice of the measurement ensemble $\mathcal{U}$ and the characteristics of the observables $O_i$. Note that the formalism can also be used to estimate nonlinear functions such as ${\rm Tr}[\rho\otimes \rho O]$~\cite{huang_predicting_2020}. Classical shadows offer an advantage with respect to maximum-likelihood and Bayesian techniques in certain tasks described in Ref.~\cite{lukens_bayesian_2021}.
Finally, if the target functions only include Pauli observables, the formalism can further be derandomised to optimise its performance~\cite{huang_efficient_2021}.
Examples for applications of classical shadows are the efficient measurement of the entanglement entropy~\cite{elben_mixed-state_2020}, or the estimation of the quantum Fisher information~\cite{rath_quantum_2021}.

\subsection{Optimising qubit readout}

Performing \gls{qst} is greatly facilitated by the availability of high-fidelity single-shot readout for each qubit state. A single-shot measurement typically consists of a time-resolved signal which is processed to produce histograms relating to the qubit being prepared in the ground or excited state, and a threshold of the readout value is chosen to resolve the two states~\cite{neumann2010single,myerson2008high,vamivakas2010observation,elzerman2004single}.

Stochastic relaxation of the excited state causes an asymmetry in readout histograms that decreases readout fidelity by threshold classification. Clustering methods have been used to identify and discard such relaxation signals and increase fidelity in superconducting qubits~\cite{magesan2015machine}. However, while still increasing overall fidelity, similar methods have produced outlier results of lower quality than thresholding repetitive readouts ~\cite{liu2020repetitive}. Non-linear Bayesian filters have improved on threshold fidelities by considering the full time-resolved readout signal while accounting for relaxation~\cite{gambetta2007protocols} and stochastic turn-on times in spin-to-charge conversion readout~\cite{d2014optimal}.
\glspl{nn} have improved on threshold methods in diamond NV centres, which lack single-shot readout at room temperature ~\cite{liu2020repetitive}, and \gls{nn} classifiers trained on synthetic time-resolved data have surpassed the performance of Bayesian filters in quantum-dot spin qubit readout~\cite{struck2021robust}. Multi-qubit states have been classified using \glspl{nn} in trapped-ion qubits, where time-binned multi-channel data has provided the highest fidelity~\cite{seif2018machine}. \glspl{rnn} have been used to predict time-evolving qubit states from experimental noisy measurement traces \cite{flurin2020using}.

While the availability of single-shot readout and projective quantum measurements facilitates learning of quantum systems, it is not necessarily a prerequisite and averaged readout can be used, in combination for example with Bayesian inference \cite{dinani_bayesian_2019}.

\section{Learning quantum dynamics}\label{sec_dynamics}
The reconstruction of quantum evolution is important for establishing, for instance, channel fidelity in quantum communication, gate fidelity in quantum computing and optimal parameter encoding for sensing applications. Following the discussion of \gls{qst}, we will first review assumption-free methods, and then move to methods that introduce specific dynamical models to simplify the characterisation. We will then discuss applications to quantum sensing and imaging.

\subsection{Learning dynamics with Quantum Process Tomography}
\label{sec:qpt}

The task of fully reconstructing the dynamics of a quantum system driven by an underlying, unknown process is called \gls{qpt}\cite{Chuang_1997,mohseni_direct_2006}.
In a typical setting, \gls{qpt} results in the estimation of a given quantum process, by applying the latter to a set of fiduciary quantum states.
The resulting output states are reconstructed via \gls{qst}, allowing one to identify how the initial states evolved under the process. More formally, one can reconstruct the quantum channel $\Lambda$
that describes
the quantum process according to
$ \rho_{\textrm{out},i} = \Lambda\left[ \rho_{\textrm{in},i}\right],
    \label{eq:QPT_mapping} $
where $\{\rho_{\textrm{in},i}\}_i$ is an arbitrary set of input states, and each $\rho_{\textrm{in},i}$ is associated to a corresponding output state $\rho_{\textrm{out},i}$ ~\cite{Poyatos_1997, Altepeter_2003, Polino_2020}.

In the same way as \gls{qst}, \gls{qpt} requires a number of measurements and classical post--processing that scale exponentially with the size of the quantum system under study~\cite{mohseni_quantum-process_2008}. Such punishing scaling afflicts, for example,
methods based on {\bf maximum likelihood inference}~\cite{kosut2004optimal}, which additionally show a high sensitivity to errors in initial states, gates and measurements\cite{merkel2013self}.

Different methods have been explored to overcome these limitations. An example is {\bf randomised benchmarking}~\cite{Knill_2008} which estimates the error rate (or fidelity) of a quantum process from a sequence of random repetitions of the process \cite{Epstein_2014,Rodionov_2014,Granade_2015, Claes_2021}. However, randomised benchmarking does not reconstruct the process matrix $\Lambda$ and thus does not deliver a fully general characterisation of the quantum process.
For processes $\Lambda$ that can be represented as a sparse superoperator, {\bf compressed sensing} (CS) methods can be employed to reduce reconstruction costs ~\cite{Shabani_2011,kosut_quantum_2009,Rodionov_2014}, cf. Sec.~\ref{sec:Efficient_QST}.

Alternative approaches exploit more sophisticated methodologies. For example, building on methods introduced for \gls{qst} (see Sec.~\ref{sec:Efficient_QST}), tensor networks
can lead to drastically reduced resource requirements~\cite{torlai_quantum_2020, gazit_quantum_2019, bennink_quantum_2019}.
\gls{gan}-based approximations of the superoperator $\Lambda$ have also been proposed as an efficient method for \gls{qpt}~\cite{braccia2021quantum}.
Other methods exploit \glspl{nn} to generalise \gls{qpt} to the characterisation of time dependent  spin systems~\cite{Han_2021}, while yet another class reconstructs a unitary quantum process by inverting the dynamics using a variational algorithm~\cite{carolan_variational_2020,xue_variational_2021}. \glspl{rnn} have recently been shown to be useful in learning the non-equilibrium dynamics of a many-body quantum system from its nonlinear response under random driving\cite{mohseni2021deep}.

\subsection{Reconstructing the generators of Hamiltonian quantum dynamics}
\label{sec:gen_learning}

Reconstructing the general superoperator $\Lambda$ is resource-demanding and one might not have access to the full set of observables required for full \gls{qpt}.
 A commonly-used approach is the reconstruction of a specific dynamical model described by {\bf generators}, e.g. a Hamiltonian model for a unitary evolution with a small number of tunable control parameters. These approaches leverage prior information about the  system to simplify the reconstruction.

A first class of methods relies on the measurement of time traces of measurable observables~\cite{di_franco_hamiltonian_2009, cole_PRA_2005, devitt_PRA_2006}.
Hamiltonian parameters for a few qubits can be retrieved by fitting these time traces or by using {\bf Fourier methods}.
More sophisticated approaches rely on using {\bf linear systems theory}, such as the eigenstate realisation algorithm~\cite{zhang_quantum_2014, sone_cappellaro_2017}. Here, after decomposing the Hamiltonian into Hermitian operators, the evolution can be described by a linear differential equation.
The algorithm then follows linear systems theory, by adopting a \textit{system realisation} from experimental data attained at regular time intervals, leading to a transfer function mapping input states to observables, from which one obtains the unknown parameters. More recent versions further refine these methods by extracting the eigenfrequencies of $H$ from a complex time domain measurement and then recovering the eigenvectors of the Hamiltonian via constrained manifold optimisation~\cite{hangleiter2021precise}.

A second class of methods employs (approximate) {\bf Bayesian inference} (see Fig \ref{fig:fig2_bayesian}b). Computationally efficient  Bayesian tomography, for example, has been used to characterise logic gates for spin qubits in silicon \cite{evans2022fast}.
A generation of Bayesian methods known as \gls{qhl} infers a set of unknown parameters $\mathbf{x}$ for a specific parametrisation $H(\mathbf{x},\boldsymbol{\tau})$~\cite{granade_robust_2012, wiebe_hamiltonian_2014, wiebe_quantum_2014}.
Here, $\boldsymbol{\tau}$ represents a set of tunable control parameters that typically is just the evolution time $t$. It is assumed that the specific parametrisation $H(\mathbf{x},\boldsymbol{\tau})$ offers a much more efficient reconstruction, as the number of parameters $\mathbf{x}$ is usually much smaller than the number of parameters necessary to describe the most general $\Lambda$ or $H$.
The implementation of \gls{qhl} requires estimating the likelihood $P (\mu | \mathbf{x},\boldsymbol{\tau})$ for each of the discrete outcomes $\mu$ of a projective measurement, and using Bayes rule to update the belief on $\textbf{x}$, encoded in the posterior distribution (see Box~\ref{sec:bayesian_box}).
To extract $P (\mu | \mathbf{x},\boldsymbol{\tau})$, QHL invokes access to a trusted simulator that can implement a controlled evolution by $H(\mathbf{x},\boldsymbol{\tau})$ for different combinations of $(\mathbf{x},\boldsymbol{\tau})$, prepare fiducial initial states, and perform projective measurements on the evolved states .

For few-qubit systems~\cite{granade_robust_2012, granade_qinfer_2017, stenberg_resonant_2014}, the likelihood may be  evaluated on a classical computer for all necessary combinations of $(\mathbf{x},\boldsymbol{\tau})$. This becomes rapidly unfeasible for larger quantum systems and the quantum nature of the simulator becomes crucial~\cite{wiebe_hamiltonian_2014, wiebe_quantum_2014, wiebe_bootstrapping_2015}.
\gls{qhl} has been successfully demonstrated in different experimental platforms, for various tasks, such as characterising NV centres in diamond~\cite{wang_experimental_2017, hou_2019_11qubit, gentile_learning_2021} or quantum sensing of magnetic fields~\cite{hincks_statistical_2018, santagati_magnetic-field_2019, joas_online_2021}, and can potentially be adapted for quantum control~\cite{wittler_integrated_2021}.
\gls{qhl} is more robust than e.g. Fourier analysis~\cite{schirmer2015ubiquitous, santagati_magnetic-field_2019} due to its Bayesian nature and the use of an adaptive choice of measurement settings according to the cumulative knowledge inferred about the system.

One of the main drawbacks of \gls{qhl} is that it requires the knowledge of the parametrisation $H(\mathbf{x})$. Recently, this limitation has been addressed by the use of \glspl{qmla}~\cite{gentile_learning_2021, flynn_2021_qmla} (Fig \ref{fig:fig2_bayesian}c), where the artificial agent constructs different candidate Hamiltonian models $H_i (\mathbf{x})$ starting from a number of simple, generic terms $h_{\nu}$ (e.g. Pauli matrices). The latter are selected and combined via tree searches or genetic algorithms to generate new candidate models. After a parameter training via \gls{qhl}, the different $H_i (\mathbf{x})$ are systematically compared against remaining candidates by metrics $\mathcal{B}$ such as Bayes factors or modified Elo-ratings~\cite{gentile_learning_2021, flynn_2021_qmla}, and the least performing instances are discarded (see Fig. \ref{fig:fig2_bayesian}c).

The task of Hamiltonian learning has also been addressed by {\bf  deep learning methods}. For example, \glspl{nn} were used to recover Hamiltonians from local measurements of ground states\cite{xin_local-measurement-based_2019}, and \glspl{rnn} can learn time-dependent target Hamiltonians from time traces of single-qubit measurements without knowledge of the ground states\cite{che_physrevresearch_2021}.
However, the training of \glspl{nn} is notoriously time-consuming. A possibility to circumvent this problem is to include inductive bias coming from the known physical laws describing the system.
One option consists of Neural Ordinary Differential Equations (\glspl{node}) that include the system's differential equations~\cite{chen2018advances} or the system's Lagrangian function~\cite{cranmer2020lagrangian} in the structure of the \gls{nn}, see Fig.~\ref{fig:NNs}(d). \glspl{node} have been applied to the optimal control of a qubit that is modeled as a closed system~\cite{schafer2020differentiable} and as a monitored (and thus stochastic) system~\cite{schafer2021control}. A second example are Physics-Inspired Neural Networks (\glspl{pinn}) which directly model the solution of the system's differential equations by including the latter in the cost function~\cite{Raissi2019pinn}, see Box~\ref{sec_NNbox}. Several software suites readily adaptable or devoted specifically to \glspl{pinn} have been developed both in research and industrial environments~\cite{rackauckas2017julia_pde, lu2021deepxde, haghighat2021sciann, hennigh2021nvidia}.
Similar to the \gls{qhl} methods above, the availability of the system's dynamical equations crucially reduces the complexity of the training whilst preserving excellent performance. Recent improvements extend \glspl{pinn} to the optimisation of several competing \gls{pinn} models~\cite{Both2021deepmod, Heim2021qmod}, and classical \glspl{pinn} may be supported by Q\glspl{nn}, i.e., by employing parameterised quantum circuits~\cite{Benedetti2019qnn, Goto2021qnn_ufa}. These circuits would then act as a quantum simulator, possibly surpassing limitations in the expressivity of classical \glspl{nn} of feasible size~\cite{McClean2018barren, Kyriienko2021dqc}.
To this date, the application of \glspl{pinn} to quantum systems is in its very infancy,
with early applications exploring simple instances of the Schr{\"o}dinger equation~\cite{Raissi2019pinn, bu2020compet_pinn} and the possibility to control dynamically-corrected quantum gates~\cite{gungordu2020pinngates}.
A final example are 'gray-box' approaches ~\cite{peruzzo_QSaT_2020, peruzzo_greybox_2022} that combine a \gls{nn} as a 'black box' to learn the Hamiltonian with a physically-understandable 'white box' embedding the rules of quantum mechanics, such as state evolution. This approach is able to address the uncertainty in the Hamiltonian model, distortions caused by undesired macroscopic dynamics and imperfect measurements.

If the target Hamiltonian $H$ is local and one is provided with {\bf knowledge of a single (approximate) eigenstate} of $H$, it was shown  that $H$ can be reconstructed using resources that scale polynomially with the number of qubits \cite{qi_determining_2019, chertkov2018inverse, greiter2018identify}.
The technique consists of  manipulating a covariance matrix of expectation values of Hermitian operators that span the space of local Hamiltonians.
This approach can be extended to generic mixed states commuting with $H$~\cite{bairey_learning_2019},
and to situations where only local measurements of limited regions of the full systems are available \cite{zhu2019reconstructing}.
These methods, however, are still sensitive to noise \cite{qi_determining_2019, bairey_learning_2019}.

\subsection{Reconstructing open quantum system dynamics}

When the dynamics of a quantum system of interest is not sufficiently isolated from its environment, the Hamiltonian description fails and the system has to be treated as an {\bf open quantum system}\cite{breuer2002theory,leggett1987}. For a memory-less environment, the system's dynamics is Markovian and the evolution of the quantum state $\rho$ can be described by the \gls{gksl} master equation~\cite{gorini1976completely,lindblad1976generators}
\begin{equation}
	 \dot{\rho} = -i[H,\rho] + \sum_j \left(L_j \rho L_j^\dagger - \{L_j^\dagger L_j,\rho\}/2\right),
	\label{eq:lindblad}
\end{equation}
where $\{A,B\}=AB+BA$, $H$ is the Hamiltonian, and $L_k$ are the Lindblad operators that describe the dissipative process. The reconstruction of $H$ and $L_k$ from measurement data
has been dubbed {\bf Lindblad tomography}~\cite{samach2022lindblad}, where an algorithm based on maximum likelihood estimators was presented. Alternatively, the generators can be reconstructed from the measurement of time traces and linear system realisation theory~\cite{zhang_identification_2015}. Local Markovian dynamics can also be recovered via local measurements when steady states can be prepared \cite{bairey202open}, adopting techniques similar to those outlined for Hamiltonian reconstruction, when knowledge of approximate eigenstates is available.

Whilst often providing an excellent description of the quantum dynamics, the assumptions resulting in a Markovian evolution are not fulfilled in general~\cite{breuer2002theory,rivas2011open, li2018concepts}.
In non-Markovian dynamics, the state at time $t+\mathrm{d}t$,  $\rho_{t+\mathrm{d}t}$, depends not only on $\rho_{t}$ but also on the system's history at earlier times.
This evolution can be expressed by the master equation~\cite{breuer2002theory}
$\dot\rho_t = \int_0^t \mathrm{d}s \mathcal K(t,s) \rho_s$ with a time-nonlocal superoperator $\mathcal K$.
Alternatively, we can employ a process tensor formalism~\cite{pollock2018non}, where the time-discretised unitary  evolution operator of the system is interleaved with environmental influence interventions from a 'quantum-comb'-like process tensor.
Whilst the dimension of this process tensor scales exponentially with the number of time steps, often, e.g.~when memory effects are short-range, it is possible to express the process tensor in compressed matrix operator form (c.f. Eq.~\eqref{eq:mps}), which can be efficiently constructed from knowledge of the underlying microscopic model~\cite{jorgensenPT_2019, cygorek2021numerically} or reconstructed from time measurements~\cite{pollock2018non,luchnikov_machine_2020}. Such {\bf process tensor tomography} generalises \gls{qpt}, thereby providing access to multi-time correlations, and it has recently been shown to be compatible with NISQ devices\cite{white2022-PTT}.
Alternatively, methods based on \glspl{nn} can be employed to reconstruct non-Markovian quantum dynamics
\cite{banchi2018modelling, krastanov_unboxing_2020}. Generally, these approaches entail reduced physical insight, however, in some cases a degree of interpretability can be preserved, e.g., by rendering a \gls{gksl} master equation \eqref{eq:lindblad}
non-Markovian through effective time-dependent Lindblad operators that depend on the entire history.
In one such example, the matrix elements of $L_k$ were expressed using \glspl{rnn}~\cite{banchi2018modelling}, i.e. neural networks of a type that are routinely applied  to model long-range memory effects.

\subsection{Parameter estimation with quantum sensors}
Quantum sensors\cite{DegenRMP2017, PezzeRMP2018, GLM2011} are devices used to estimate parameters -- such as magnetic fields, external forces, accelerations,  etc. -- that modify (e.g. via a dynamical parameter encoding) a quantum state.
Quantum systems provide a key advantage in terms of their small size (resulting in high spatial resolution).
Furthermore, quantum effects such as coherence, squeezing and entanglement can increase the sensitivity of the device~\cite{PezzeRMP2018, GLM2011}.
The optimal uncertainty for the unbiased estimation of a parameter $\theta$, based on $m$ measurement results, can be quantified by the quantum Cramer-Rao bound~\cite{helstrom_quantum_1976, BraunsteinPRL1994}
\begin{equation} \label{QFI}
    (\Delta \theta)^2 = \frac{1}{m F_Q(\theta)},
\end{equation}
where $F_Q$ is the \gls{qfi}, depending on the sensor's output state $\rho_\theta$.
Equation (\ref{QFI}) is understood as the sensitivity achievable with an optimal estimator and using optimal measurement observables~\cite{BraunsteinPRL1994}.
In a Bayesian setting (see Box \ref{sec:bayesian_box}), Eq. (\ref{QFI}) provides the asymptotic posterior variance using optimal measurements.
It is important to emphasise that the frequentist Cramer-Rao bound Eq. (\ref{QFI}) does not hold for Bayesian inference and, thus, Bayesian bounds must be used instead~\cite{LiENTROPY2018}.
Recently, it has been shown that \glspl{nn} can then be trained to construct estimators~\cite{cimini_calibration_2019, cimini_calibration_2021, NolanQST2021} or Bayesian posterior distributions~\cite{nolan_machine_2020} based on supervised learning techniques and using limited calibration data.
In both cases, a well-calibrated network can be used for post-processing data that saturate Eq.~(\ref{QFI}) for a sufficiently large number of measurements.
These \gls{nn}-based estimation methods are valuable when one is lacking simple models for the output probability distribution of the sensor, e.g. when the number of possible measurement events is large.

A notable result in quantum sensing is the relation between uncertainty and entanglement~\cite{PezzeRMP2018, TothJPA2014}: separable states of $N$ qubits that undergo a collective spin rotation (as common in many applications) can achieve, at best~\cite{PezzePRL2009}, a \gls{qfi} $F_Q=N$.
Entanglement is necessary~\cite{PezzePRL2009} for $F_Q>N$, while genuine multipartite entanglement is necessary~\cite{HyllusPRA2012, TothPRA2012} to achieve the ultimate limit $F_Q=N^2$.
Interestingly, multipartite entanglement witnessed by the \gls{qfi}~\cite{HyllusPRA2012, TothPRA2012} is found in a variety of many-body phenomena such as quantum phase transitions, quantum chaos, quenches, random states, and so on.

In the case of the joint estimation of $d$ parameters $\theta_1, ..., \theta_d$, the $d\times d$ covariance matrix of estimators is bounded the inverse (when it exists) of the \gls{qfi} matrix \cite{helstrom_quantum_1976, AlbarelliPLA2020}.
In general, when estimating multiple parameters, there is no optimal observable with which the quantum Cramer-Rao bound can be saturated.
This problem is due to non-commutativity of the different parameter encoding transformations~\cite{helstrom_quantum_1976, AlbarelliPLA2020} and the more involved Holevo-Cramer-Rao bound must be used instead \cite{RafalPLA2020}.
For commuting encoding transformations, the interplay between mode and particle entanglement and the \gls{qfi} matrix has been discussed in Ref. \cite{GessnerPRL2018} and experimentally investigated with optical qubits\cite{LiuNATPHOT2020}.

\subsection{Quantum Imaging, Target Detection and Pattern Recognition}

As in quantum sensing, quantum-enhanced detection schemes use a combination of entangled states of light and carefully designed measurements to beat the classical limits. However, unlike metrological applications where the task is typically to estimate a continuous parameter, here the main task is to estimate a discrete variable, such as the binary variable describing the presence or absence of a target or the pixel values in the images.
Because of the discrete nature of the problem, a quantum advantage cannot be quantified via the quantum Cramer-Rao boundEq. (\ref{QFI}), and alternative strategies have been proposed. A common approach is to first define a problem-dependent cost function and then compare the performance of entangled sensors with strategies based on coherent states only.

Since the field is vast, here we limit ourselves to \gls{nn}-inspired methods and refer to existing
reviews~\cite{moreau2019imaging,pirandola2021quantum2} for a more extensive treatment.
Entangled sensors have been used to define a quantum support vector machine able to more accurately classify the presence of a target \cite{zhuang2019physical}. Moreover, entangled probes can be used to read binary images with enhanced precision and perform more accurate image classification \cite{banchi2020quantum}.
While for full imaging each pixel must be accurately reconstructed, for pattern recognition errors are tolerated and less precise detectors are allowed. This intuitive idea was formalised in \cite{banchi2020quantum}  by showing that the error with optimal detectors decreases exponentially with the number of bits that must be flipped to switch class.
Finally, quantum imaging can be enhanced by deep learning techniques \cite{li2021fast,picard2019deep,harney2021ultimate}, in order to increase the fidelity, discover deeper structure in data and overcome errors due to shot noise and background noise.

\section{Learning quantum measurements}\label{sec_measurements}
\label{sec:qmeasure}

The characterisation of detectors is commonly given in terms of quantum efficiency, linearity, rate of dark counts, and spectral and temporal response.
However, these aspects represent only an approximation of the actual operation of a detector and can introduce systematic errors that may strongly affect high-precision measurements or investigations of quantum effects. \gls{qdt}~\cite{LuisPRL1999,FiurasekPRA2001} aims at reconstructing quantum measurement devices without any prior information or approximations.
This characterization plays a key role in any quantum architecture.

Realistic (noisy) detectors are non-projective and thus
\gls{qdt} consists in the tomographic reconstruction of a set of \gls{povm} operators $E_\mu$~\cite{LuisPRL1999,FiurasekPRA2001,DArianoPRL2004}.
These operators
describe the probability of a generic measurement detection event $\mu$ according to the Born rule
\begin{equation} \label{QDT}
    \Pr(\mu) = {\rm Tr}[\rho E_\mu],
\end{equation}
given a probe state $\rho$.
The standard \gls{qdt} approach consists of inverting Eq.~(\ref{QDT}) using the experimentally-sampled probability distribution and for a suitably chosen set of probe states that must be tomographically complete, i.e., span the Hilbert subspace where the POVM elements are defined.
As a simple example, we can consider an ideal (von Neumann) projective measurement $\Pr_{\rm id}(\nu) = \langle \nu \vert \rho \vert \nu \rangle$, where $\{ \vert \nu \rangle \}$ is a set of orthogonal states, and model a noisy detection by a Markov mapping $V\geq 0$ such that $\Pr_V(\mu) = \sum_\nu V_{\mu\nu} \Pr_{\rm id}(\nu)$ is the probability of a detection event, depending on the specific choice of $V$.
As $V_{\mu\nu}$ describes the probability to observe the result $\mu$ when $\nu$ should be ideally observed, it includes the detector's imperfections, and satisfies the normalization condition $\sum_\mu V_{\mu\nu}=1$ for all $\nu$.
In this case, given the observed probabilities $\Pr(\mu)$, \gls{qdt} corresponds to finding the matrix $V$ that minimises the distinguishability between the probability distributions $\Pr(\mu)$ and $\Pr_V(\mu)$ (e.g. quantified by the Kullback-Leibler divergence or the fidelity).
Such an optimisation can be obtained via gradient descent~\cite{hetzel_tomography_2022} or a maximum likelihood method.
The corresponding POVM in Eq.~(\ref{QDT}) reads $E_\mu = \sum_\nu V_{\mu\nu} \vert \nu \rangle \langle \nu \vert$.

Different detector-tomography techniques have been developed and used experimentally, with a main application being the characterisation of optical photocounting~\cite{LundeenNATPHYS2009, DAuriaPRL2011, BridaNJP2012} and homodyne~\cite{ZhangNATPHOT2012, GrandiNJP2017} detection, and, more recently, qubit readout in quantum  computing  machines~\cite{ChenPRA2019}.
With superconducting single photon detectors, \gls{qdt} has provided a valuable tool to discriminate among different models to explain the basic physics underpinning the detection mechanism \cite{RenemaPRL2014}.
Usual \gls{qdt} methods require the precise calibration of probe states that, in turn, demands the precise knowledge of the measurement device, thus possibly inducing systematic errors~\cite{MogilevtsevNJP2012, KeithPRA2018}.
Self-characterising \gls{qdt} techniques that do not rely on precisely-calibrated probe states have been demonstrated for single optical qubits~\cite{ZhangPRL2020}.

\section{Optimising quantum experiments}\label{sec_optimisation}
In this section, we briefly review a few selected applications of classical learning for the optimisation of quantum experiments.

\subsection {Adaptive methods for quantum sensing}
Quantum sensor can be optimized by harnessing {\bf adaptive protocols}: these, that compute the optimal experimental settings from the current knowledge about the unknown parameter being estimated, following data acquisition.
Indeed, while the optimal setting of the sensor depends, in general, on the unknown parameter being estimated, adaptive strategies can converge, asymptotically in the number of measurements, to the optimal configuration.
Bayesian methods (see Box \ref{sec:bayesian_box}) are well suited for adaptive optimisation~\cite{HigginsNATURE2007, FidererPRXQuantum2021} (Fig \ref{fig:fig4_optimisation}a and b).
Several adaptive protocols have focused on the optimisation of measurements \cite{BerryPRL2000, HigginsNATURE2007}, including numerical techniques such as particle swarm~\cite{ hentschel_efficient_2011} and differential evolution algorithms~\cite{lovett_differential_2013}.
Recently, \gls{rl} has been employed to discover the whole sequence of optimal adaptive measurements of a qubit in the Bayesian framework \cite{FidererPRXQuantum2021}.
The optimization of probe states, e.g. in order to extend the bandwidth of entanglement-enhanced sensitivity, has been considered using analytical methods \cite{PezzePRL2020, PezzePRXQuantum2021} as well as
variational~\cite{KaubrueggerPRL2019, KaubrueggerPRX2021} and \gls{nn} strategies~\cite{HainePRL2020}.
Besides the optimisation of measurements and probe states, another possibility is to modify parameter encoding by adding control Hamiltonians\cite{ pang_optimal_2017,xu_generalizable_2019}, optimised efficiently via \gls{rl}.

\begin{figure*}[ht]
\centering
\includegraphics[width=.95\linewidth]{Fig4.pdf}
\caption{
{\bf Optimising quantum experiments} {\bf (a)} Example of adaptive experimental control, through Bayesian inference. After each measurement, the outcome is used to update the current probability distribution for the parameters of interest. This probability distribution is then used to compute optimal settings for the next measurement, through ad-hoc heuristics, the optimisation of statistical quantities (variance, Fisher information, Kullback-Leibler divergence, etc) {\bf (b)} Adaptive Bayesian optimisation has been used, for example, to optimise magnetic sensing using the single electronic spin associated to an NV centre in diamond. In the plot, from T. Joas et al \cite{joas_online_2021}, as estimation epochs increase, the magnetic field value converges to the ground truth (solid lines) and the standard deviation (shaded areas) decrease. {\bf (c)} Schematic feedback loop optimised via \gls{rl}. Measurement results are fed into an agent that decides on the next action to apply to the quantum experiment. {\bf (d)} Different scenarios for optimised quantum control; time evolution, with control pulses as black arrows, measurements as orange arrows, agent in brown, and the quantum experiments' coherent evolution intervals in green. {\bf (e)} Adaptive Bayesian inference and \gls{rl} have been applied to or suggested for several experimental quantum systems, such as NV centres ~\cite{bonato_optimized_2016, santagati_magnetic-field_2019, dushenko_sequential_prappl2020, joas_online_2021, mcmichael_sequential_2021, childress2022}, quantum dots \cite{lennon2019efficiently,nguyen2021deep}, cavity-qubit systems \cite{sivak2022model,porotti2021deep} and multi-qubit systems with gates applied as actions and subject to projective measurements \cite{fosel_reinforcement_2018}}.

\label{fig:fig4_optimisation}
\end{figure*}

Experimentally, adaptive techniques have been employed, for example, to improve the sensitivity of optical phase sensing realizing a phase estimation protocol \cite{HigginsNATURE2007}, or in quantum sensors based on the single spin associated with the nitrogen-vacancy (NV) centre in diamond, a system widely used for nanoscale magnetic mapping and magnetic resonance~\cite{bonato_optimized_2016, santagati_magnetic-field_2019, dushenko_sequential_prappl2020, joas_online_2021, mcmichael_sequential_2021, childress2022}.
Due to the time constraints, a key consideration for online sensing is the cost of the processing procedure: simplified near-optimal methods might perform better than optimal but computationally-intensive ones. This is especially important in the tracking of time dependent magnetic fields in DC magnetometry ~\cite{bonato_tracking_2017, hincks_statistical_2018, santagati_magnetic-field_2019}.

\subsection{Optimal control over the environment}
Reconstructing the dynamics of a quantum system presents opportunities to establish a degree of control over the environment, for example with the goal of reducing the decoherence of the system. \gls{qhl} has been used to suppress decoherence for a single spin qubit by a real-time compensation of  nuclear-spin bath fluctuations~\cite{shulman_suppressing_2014} and classical noise~\cite{mavadia_prediction_2017}.
If the environment can be described by a quantum system, the learning process itself can be considered a control tool because the back-action of a quantum measurement perturbs the quantum state, for example by projecting the system to an eigenstate of the measurement~\cite{cappellaro_spin-bath_2012}.
Even though the random measurement outcomes result in random quantum states of the system, feedback can be used to make the control system deterministic~\cite{blok_manipulating_2014}. Protocols have been suggested to extend a qubit's coherence by sequential Hamiltonian learning with real-time feedback from the measurement outcomes~\cite{Scerri_2020}.

A similar idea is at the basis of quantum error correction, which is a general approach to counteract noise in quantum computation and quantum communication\cite{nielsen_quantum_2011}.
More formally, given a general noisy evolution $\Lambda$, quantum error correction consists in finding a recovery channel $\mathcal{R}$ such that
\begin{equation}
    \mathcal{R}[\Lambda[\rho_{S}]] =  \rho_{S},
\end{equation}
where $\rho_{S}$ is defined in a subspace $\mathcal{H}_S$ of the total Hilbert space.
For applications in quantum sensing~\cite{KesslerPRL2014, ArradPRL2014, DurPRL2014, ZhouNATCOMM2018, GoreckiQUANTUM2020}, it is further required that the recovery operation $\mathcal{R}$ is such that $\mathcal{R}[\Lambda[\rho_{S}(\theta)]]$ follows a noiseless parameter-encoding dynamics in the subspace $\mathcal{H}_S$ including quantum states with finite (possibly large) quantum Fisher information.

\subsection{Optimal control of a quantum system}

Other applications of \gls{ml} methods to quantum technologies include the design of quantum gates~\cite{Innocenti_2020, Gao2020} or the identification and design of circuits for quantum error correction~\cite{valenti_hamiltonian_2019}.
In particular, model-free \gls{rl} is a powerful general approach to learn control strategies, where an agent (often implemented via a \gls{nn}) observes and controls an "environment", which might be a quantum device (see Fig.\ref{fig:fig4_optimisation}c). The dynamics of the environment need not be known in advance: the agent will implicitly learn an approximate model while trying to find a good feedback control strategy. Overall, the agent tries to optimise a reward (e.g. defined in terms of final state fidelity for the case of state preparation). In so-called policy gradient approaches, this is done by implementing a parametrised stochastic policy $\pi_\theta(a|s)$ -- the probability of an action $a$ given an observed state $s$ -- and changing by gradient ascent the probabilities in any action sequence to maximise the average cumulative reward (see e.g. Ref \cite{marquardt2021machine} for more details).

In theoretical studies, RL has been shown to discover feedback strategies for quantum error correction  \cite{fosel_reinforcement_2018}, obtain qubit control pulses for state preparation \cite{bukov2018reinforcement,niu2019universal}, and solve many other tasks (see Fig.\ref{fig:fig4_optimisation}d and e). It is beginning to be implemented on experimental platforms, e.g. for superconducting qubit gate synthesis \cite{baum2021experimental} and also for adaptive characterisation of quantum systems \cite{nguyen2021deep}. A particular challenge consists in extracting the reward reliably from available experimental measurements and to make sure that the agent can be trained in a truly model-free way without any assumptions \cite{sivak2022model}.

\subsection{Optimal design of experimental quantum setups}

In addition to optimising control strategies for given experimental setups, one can also pursue the goal of optimising the setup itself. This may involve varying the components and how they are connected within the setup. In a quantum-optical setup, these might be linear and nonlinear components. Whole setups in this domain have been automatically discovered, optimising some desired property like the entanglement of  multi-photon states produced at the output\cite{Kottmann_2021, krenn_computer-inspired_2020}.
It has then been possible to extract motifs from the generated setups and identify them as useful building blocks that implement some novel ideas: see Ref.\cite{krenn_computer-inspired_2020} for an overview of experimental implementations of optimised setup designs

\section{Discussion and Perspectives}

The application of classical algorithms and methods to quantum technologies offers possibilities to circumvent some of the obstacles in characterisation, calibration and validation posed by the exponential complexity of quantum systems and the intrinsic probabilistic nature of quantum measurements.
In this review, we have focused on classical data-processing of measurement outcomes of quantum experiments and their optimisation towards a variety of tasks, ranging from learning states to controlling experimental setups.
The fast progress of this field relies on the possibility to directly adapt efficient numerical methods already developed in computer science and statistics to solve demanding problems in quantum physics, such as the characterisation of quantum states, the identification of quantum phases of matter and entanglement witnesses, the reconstruction of quantum dynamics, the optimisation of parameters and operations, etc.

Most adaptive experiments to date have been harnessing Bayesian methods, due to their simple practical implementation. Bayesian inference, however, relies on the availability of explicit fitting models, precisely calibrated to the specific experimental setting. When such models are not available, or to remove the requirement of tedious calibration, neural networks can provide a better option,because of their superior expressivity. We expect that in the future, a growing number of experimental implementations will employ neural networks to model "black-box" quantum sensors or systems.
This also holds for online parameter optimisation in adaptive experiments: while most experiments so far have relied on near-optimal simple analytical heuristics, deep \gls{rl} has been suggested \cite{FidererPRXQuantum2021} as a "system-agnostic" procedure to generate optimised ready-to-use heuristics. We expect that \gls{nn}-based inference and \gls{rl} will be increasingly adopted by experiments, removing the need for manual derivation of experimental-design heuristics.

More generally, \gls{rl} is a very promising approach in discovering from scratch feedback-control strategies for quantum devices, for goals ranging from state and device characterisation to error correction. While it works successfully for some tasks in simulations and also in first experiments, there are still multiple demanding challenges to be solved in this domain. Among them the need for the network agent to find simultaneously both a good control strategy and an interpretation of the noisy measurement data, the technical hurdles in experimentally implementing real-time feedback using neural networks, and the construction of good reward functions that help guide the optimisation while being accessible from experimental data.

Further opportunities come from developments of quantum computing hardware. While a classical computer processes the classical outcomes of quantum measurements, a quantum computer can directly handle "raw" quantum states from an experiment. The main advantage of this approach is that a quantum computer can perform joint measurements on several qubits, exploiting quantum entanglement. A recent paper \cite{google_science_2022} showed that joint quantum measurements lead to substantial advantages for learning quantum systems, as they outperform single measurements in distinguishing quantum states even when the states are uncorrelated. A challenge for learning quantum states with a quantum processor is the need for a quantum-coherent interface between the quantum system under study and the quantum processor. This will require further research in "hybrid" quantum structures, where quantum sensors, devices and systems are interfaced to a quantum computer preserving quantum coherence.

The fact that advantages in learning quantum systems can already be demonstrated by current noisy intermediate quantum devices opens up exciting possibilities, that will become more and more important as we advance towards full fault-tolerant quantum computing.

\section*{Acknowledgements}
The authors thank Chris Ferrie for useful discussions. CB is supported by the Engineering and Physical Sciences Research Council (EP/S000550/1 and EP/V053779/1), the Leverhulme Trust (RPG-2019-388) and the European Commission (QuanTELCO, grant agreement No 862721). NA acknowledges support by the Royal Society (URF/R1/191150), EPSRC Platform Grant (EP/R029229/1), the European Research Council (grant agreement 948932) and FQXi Grant Number FQXI-IAF19-01.
LB's work is supported by the
U.S. Department of Energy, Office of Science, National
Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center
(SQMS) under the contract No. DE-AC02-07CH11359, and by the INFN via the QubIT, SFT and INFN-ML projects.
VG and LP acknowledge financial support from the European Union's Horizon 2020 research and innovation programme—Qombs Project, FET Flagship on Quantum Technologies Grant No. 820419.

\onecolumngrid

\begin{mybox}[label = {sec:bayesian_box}]{Bayesian inference}

Bayesian inference is a central method in statistical analysis that prescribes how to update a degree of belief about a hypothesis (e.g. that a set of unknown parameters $\boldsymbol{\theta} =\{ \theta_1, ..., \theta_d\}$ has the specific values $\boldsymbol{\phi} =\{ \phi_1, ..., \phi_d\}$) if new observations or evidence (e.g. a sequence of measurement results $\boldsymbol{\mu}=\{\mu_1, ..., \mu_m \}$) are obtained.
It is based on {\bf Bayes' theorem}, which for parameter estimation takes the form
\begin{equation}
    P(\boldsymbol{\phi} \vert \boldsymbol{\mu}; \mathcal{C}) = \frac{P(\mathbf{\mu} \vert \boldsymbol{\phi}; \mathcal{C}) P(\boldsymbol{\phi})}{P(\boldsymbol{\mu}; \mathcal{C})},
\end{equation}
dictating how to obtain the {\bf posterior probability} $P(\boldsymbol{\phi} \vert \boldsymbol{\mu}; \mathcal{C})$ from the {\bf prior distribution} $P(\boldsymbol{\phi})$, using the likelihood $P(\boldsymbol{\mu} \vert \boldsymbol{\phi}; \mathcal{C})$ of observing $\boldsymbol{\mu}$ with measurement settings $\mathcal{C}$ if the true phase was $\boldsymbol{\phi}$, and a marginal probability $P(\boldsymbol{\mu}; \mathcal{C})$ for normalisation.
In quantum systems, $P(\boldsymbol{\mu} \vert \boldsymbol{\phi}; \mathcal{C})$ is computed using Born's rule.
Estimates and corresponding uncertainties (or confidence intervals) can be obtained directly from the Bayesian posterior: this provides a crucial advantage with respect to the frequentist approach that obtain parameter uncertainties from measurement histograms.

If the measurement results $\boldsymbol{\mu}$ are independent, Bayes' theorem can be also written recursively
in the form of a {\bf Bayesian update} when observing the measurement result $\mu_j$, see Fig.~\ref{fig:fig2_bayesian}.
The Laplace-Bernstein-von Mises theorem guarantees that, asymptotically in the number of measurements $m$ and under mild regularity conditions (such as continuity, regular derivatives and absence of periodicity), the Bayesian posterior converges to a normal distribution
\begin{equation}
    P(\boldsymbol{\phi} \vert \boldsymbol{\mu}) \to \sqrt{\frac{m \det F(\boldsymbol{\theta})}{2 \pi}} e^{-\tfrac{m}{2}(\boldsymbol{\phi} -\boldsymbol{\theta})^T\cdot F(\boldsymbol{\theta})\cdot (\boldsymbol{\phi} -\boldsymbol{\theta})},
\end{equation}
centred at $\boldsymbol{\theta}$ with a covariance given by the inverse of the Fisher Information matrix $F(\boldsymbol{\theta})$
with entries $[F(\boldsymbol{\theta})]_{ij} = \sum_{\boldsymbol{\mu}} P(\boldsymbol{\mu} \vert \boldsymbol{\theta}) \partial_{\theta_i} \log P(\boldsymbol{\mu} \vert \boldsymbol{\theta}) \partial_{\theta_j} \log P(\boldsymbol{\mu} \vert \boldsymbol{\theta})$, for a fixed measurement configuration.

A further advantage of Bayesian inference is that it allows a natural integration of {\bf optimisation techniques}~\cite{BerryPRL2000, HigginsNATURE2007, wiebe2016qpe, GebhartPRAPP2021}.
At each point in time, the current knowledge $P(\boldsymbol{\phi} \vert \boldsymbol{\mu}; \mathcal{C})$ about the parameters can be used to choose optimal experimental settings $\mathcal{C}$ for the next measurement. In cases where the optimal measurement settings depend on the (unknown) parameters $\boldsymbol{\theta}$, adaptive techniques can significantly improve the performance and the requirements of the estimation protocol.
A practical difficulty is that post-processing the Bayesian distribution requires to evaluate it on a grid of $\boldsymbol{\phi}$ values, which can be problematic for broad distributions and in multiparameter scenarios.
An efficient Bayesian inference for the estimation of multiple parameters~\cite{loredo2004bayesian, granade_practical_2016, GebhartPRAPP2021} thus requires using approximate methods, such as particle-filtering and Sequential Monte Carlo methods~\cite{delmoral2006smc, granade_robust_2012, wiebe2016qpe}, or structured filtering~\cite{granade_2016_structured} when dealing with multiple, equivalent optima.

\end{mybox}

\begin{mybox}[label = {sec_NNbox}]{Machine learning and neural networks}

The basic idea of \gls{ml} is to train a computer to solve a specific task without explicitly instructing it how to operate. Central to this approach is the availability of large amounts of data (or the possibility to synthetically generate them). \gls{ml} can be employed to address various tasks \cite{goodfellow2016deep} that can be grouped into different types.
For instance, three important \gls{ml} tasks are (i) the \textbf{classification} of data into categories, (ii) the \textbf{regression} of functions given their values on data vectors, and (iii) the \textbf{sampling} of new data vectors that are distributed similar to vectors in the given data.
A central goal of \gls{ml} algorithms is generalisability, i.e., the computer should succeed in the given task not only for the given training data, but also when new data are provided after the learning phase.
This can be monitored by dividing the available data set into training and test subsets, which are used to train and test the algorithm, respectively.

The basic building block of several modern \gls{ml} architectures is the \textbf{artificial neuron}.
These are single-output nonlinear functions $n:\mathbb{R}^n\rightarrow \mathbb{R}$ typically modeled as $n(\mathbf{x})=f(\mathbf{W}\cdot \mathbf{x}+b)$, where $f:\mathbb{R}\rightarrow \mathbb{R}$ is a fixed nonlinear (activation) function, and the weights $\mathbf{W}$ and optional biases $b$ are optimised during the training phase.
As a single neuron is not sufficient to approximate complex dependencies, multiple neurons are arranged and connected to form a \textbf{\gls{nn}}.

Widely used \gls{nn} architectures include:

\begin{wrapfigure}{r}{.25\textwidth}
\vspace{-0.5cm}

\includegraphics[width=.85\linewidth]{sketch_NN.pdf}
\end{wrapfigure}

(1) \textbf{Dense feedforward \glspl{nn}}, with neurons grouped into several layers, where the initial and final layer describe the input and output, respectively. Any neuron can be influenced by any other neuron of the previous layer, and can influence any neuron of the subsequent layer. The intermediate (hidden) layers generate the expressivity of the \gls{nn}, where the number of hidden layers and the number of neurons in the hidden layers are called the depth and the width of the \gls{nn}, respectively.

(2) Feedforward \glspl{nn} can be equipped with \textbf{convolutional layers}, in which each neuron is influenced by only a part of the previous layer by means of a convolutional filter or kernel. This filter is the same for all neurons of a layer, significantly reducing memory and training time requirements. Convolutional layers are particularly suitable to recognise and extract regular patterns.

(3) Recurrent neural networks (\textbf{\glspl{rnn}}) allow for retaining memory between the processing of subsequent inputs. This approach is often used in time-series or feedback-control scenarios.

(4) A Restricted Boltzmann Machine (\textbf{\gls{rbm}}) employs a visible layer that is connected to one or several hidden layers. The (undirected) interlayer connections represent the weights of a Gibbs distribution, such that they can be efficiently trained via Gibbs sampling. \glspl{rbm} are used to model discrete probability distributions.

(5) Generative adversarial networks (\textbf{\glspl{gan}}) consist of two separate and competing \glspl{nn}, the generator and the discriminator. The discriminator is trained to distinguish genuine data from that generated by the generator. The resulting adversarial game converges when the generator is capable of fooling the discriminator, by generating data that are indistinguishable from genuine ones.

A large enough \gls{nn} is known to be a universal function approximator \cite{Hornik1993ufa}. However, the size of the \gls{nn} should be carefully chosen as its trainability can be compromised when chosen too large, and its generalisability may also suffer in presence of a high expressivity and long training schedules (overfitting).
To train the \gls{nn}, one must choose a problem-specific cost function that can be minimised via stochastic gradient descent. The way the \gls{nn} is trained depends on the given task and is generally divided into three categories:
\begin{center}
\begin{tabular}{|p{0.3\textwidth}|p{0.3\textwidth}|p{0.3\textwidth}|}
  \hline \textbf{Supervised learning}  &  \textbf{Unsupervised learning} & \textbf{Reinforcement learning} \\ \hline
  The training data are labeled with their target values, i.e., the target function that should be learned by the \gls{nn} is known for the training data.  & The training data are not labeled and the \gls{nn} is trained to recognise any type of structure or pattern in the data.  & There is no training data, but the \gls{nn} is connected to an environment. The \gls{nn} is trained to maximise a reward that is assigned according to the \gls{nn}'s actions.   \\
  \hline
\end{tabular}
\end{center}

Finally, we want to mention the recent introductions of neural ordinary differential equations) \glspl{node} and physics-inspired neural networks (\glspl{pinn}). \glspl{node} model differential equations~\cite{chen2018advances} (or a Lagrangian function~\cite{cranmer2020lagrangian}) with a \gls{nn}. In particular, given the current state of the system, $u(\mathbf{x},t)$, as the input, the \gls{nn} outputs the right hand side of the differential equation, see Fig.~\ref{fig:NNs}(d).
On the other hand, \glspl{pinn} are trained to directly output the solution $g(\mathbf{x},t)$ of the differential equation given the coordinates $(\mathbf{x},t)$ as the input.
Here, the differential equations are included in the cost function (at arbitrarily chosen points $\{ (\mathbf{x}_l,t_l) \}_l$) which the \gls{pinn} tries to minimise.
Boundary conditions are imposed using additional terms in the cost function~\cite{Raissi2019pinn} or via stronger constraints~\cite{Moseley2021fbpinn}.

\end{mybox}

\twocolumngrid

redpajama_set_name: RedPajamaArXiv


text:
Adson Lira Nascimento
Adson Lira Nascimento's profile at Sherdog
Association: Urso Team
Pro Debut Date: 2004-03-06
Pro Record: 38-11-2
Win Finish %: 68.4%
Quality Perf. %: 51.0%
Last Ranked: 7/01/2014
#354 Lightweight
Highest Quarterly Ranking: 1/01/2013
#116 Featherweight
Last Loss Date: 3/15/2014
Last Loss: [NR] Flavio Alvaro
Quarterly Generated Historical Rankings
Rating bar represents the fighter's standing relative to the top of the division at the time
Date ↑ ↓ Rank Record Points
07/01/2014 18 #354 Lightweight 37-11-2
04/01/2014 NR #336 Lightweight 37-11-2
01/01/2014 20 #157 Featherweight 37-10-2
10/01/2013 9 #137 Featherweight 37-10-2
07/01/2012 NR #133 Featherweight 36-10-2
04/01/2012 30 #171 Lightweight 35-9-2
10/01/2011 272 #215 Lightweight 31-8-2
07/01/2011 NR #487 Lightweight 28-8-2
07/01/2009 6 #396 Lightweight 20-4-2
10/01/2006 14 #224 Lightweight 7-1-0
04/01/2006 NR #178 Lightweight 7-1-0
A fighter's rank (US) state is assigned based on the state that the fighter most frequently fought in professionally, within the past three years.
If there is a tie, the most recent state is used. The Win-Loss-Draw record for the fighter in the "Big Leagues".
The Big Leagues refer to the top organizations based on a statistical analysis we performed.
Some organizations were counted as one, due to mergers, etc. A quality performance is a fight in which the fighter fought an acceptable opponent,
and substantiated their current rating (at the time) based on the outcome. At it's most basic level, one can equate "combat age" to "dog years" in that it will mirror the basic human lifecycle.
With this computation, we expect fighters to hit their respective "peak" in the years that your typical adult "peaks" in life, somewhere between 25 and 40.
We will expect most fighters to start their descent at some point in their 40s and be far removed from their peak by 60. This is an experimental metric that attempts to reflect a fighter's recent opposition quality.
The metric goes back 1,080 days, and looks at all the fighter's opponents' W-L-D records within a window of 540 days before,
and 540 days after their respective bout(s) with the fighter.
All of the records are added up and the cumulative win percentage is presented by this metric in a "strength of schedule" format. These ranking sets have been generated by a recent version of the software,
using recent data and will not match previously published issues of the rankings,
especially since a majority of these generated sets are for dates much earlier than the site's inception.
Mixed Martial Arts Fighter

redpajama_set_name: RedPajamaCommonCrawl


PNY Technologies® is NVIDIA's Quadro® and GeForce® channel partner throughout North America, Latin America, Europe, the Middle East, Africa and India. From Building Information Management to CAE simulation, realtime rendering, or AI enhanced products or workflows, PNY will work with you to find the ideal NVIDIA GPU for your most demanding and sophisticated graphics, HPC, visualization, simulation or AI deployments.
Established in 1985, PNY Technologies®, Inc. celebrates over 30 years of business excellence as a leading supplier of NVIDIA® Graphics Cards, and a noted manufacturer of Solid State Drives, Flash Memory Cards, USB Flash Drives and Memory Upgrades.


text:
Q: Why doesn't remove() actually remove last instance of duplicates in Python list I have this code in Python where I'm trying to remove all instances of the number 1 from a list. I also know that:

*

*It's not possible to mutate a list while iterating over it.

*remove() removes the first occurrence of an item which has multiple instances of it in the list.

my_list = [2, 3, 1, 5, 8, 4, 6, 1, 1]
print(my_list)

for num in my_list:
    if num == 1:
        my_list.remove(num)

print(my_list)

The output:
[2, 3, 1, 5, 8, 4, 6, 1, 1]
[2, 3, 5, 8, 4, 6, 1]

Considering the two points I mentioned, how is it that the list changes? And more importantly, why is the last 1 not removed from the list?

A: As the commenters said, there is actually no restriction about being able to mutate the list. Take a look at this output and see if it helps.
>>> mylist = [1, 2, 3, 1, 2, 3, 1, 2, 3]
>>> for num in mylist:
...   print(num)
...   if num == 1:
...     mylist.remove(num)
...
1
3
1
3
1
3
>>> print(mylist)
[2, 3, 2, 3, 2, 3]

Do you see how it skips the 2s? It is traversing the list from index[0] to index[-1]. When you call remove() it keeps skipping to the next index point. You never removed the 1 in your example because the for loop had already iterated over that index point.

A: 1.
You actually can modify a list while iterating it:
for i in range(len(my_list)-1, -1, -1):
    if my_list[i] == 1:
        del my_list[i]

This is a nice trick that I saw sometime ago in one of the answers here in SO (I don't remember where otherwise I would add a link). By iterating the list backwards you can delete keys and still able to continue iterating indices that are lower than the one you've just deleted!

*According to the docs, remove *should remove only the first occurrence. It's doing exactly what it's supposed to do!

redpajama_set_name: RedPajamaStackExchange


This Sunday, January 15th, at 10:30 am there will be a Covenanting at Chalmers Castle-Downs United for the Chalmers Castle-Downs conregation and Edmonton Presbytery to enter into and celebrate a covenanted relationship with Rev. Christian Kakez-A-Kapend.
I encourages as many presbytery members as possible to attend this service to bring a presence from their own congregaton as the covenanting service will be held at the same time as the ministers in the presbytery will be leading worship in their own congregations and unable to attend.


text:
Advances for multiple myeloma treatments
Mayo Clinic physicians treat patients with multiple myeloma using transformative immunotherapies with the goal of both prolonging survival for each patient and finding a cure for myeloma. Hear Mayo Clinic physicians S. Vincent Rajkumar, M.D., and Yi Lin, M.D., Ph.D., hematologists for the Mayo Clinic Cancer Center in Rochester, Minnesota, give an overview of CAR-T cell therapy and immunotherapy using specific antibodies in the treatment of multiple myeloma.
Mayo Clinic has been involved in the study and trial of both existing and emerging immunotherapies for multiple myeloma.
immunotherapy is the way of the future for multiple myeloma and immuno therapy for myeloma comes in different ways. We have cellular therapy where we use the patient's own T cells, we harvest the T cells, modify them and re inject them. And now these T cells are trained to fight the myeloma and many different parties products are being tested at mayo car T therapy or chimeric antigen receptor T cell therapy has been so transformative really, you know in a very short time now people scientists, you know very smart colleagues have been doing research in this in decades but in terms of recognizing that having a car construct that has clinical activities and then translating it very quickly. A multi center studies and getting it to the FDA for approval and we anticipate to FDA approved party product to use in standard of care clinic Myeloma cells express a marker called B cell maturation antigen or B. C. M. A. And so that's what these carty's are targeting. There is a second B. L. A filed for Cartier and multiple myeloma with the F. D. A. And Mayo clinic participated in that registration study. And so we anticipate that would also be reviewed sometime later this year and potentially be this a second carty product in multiple myeloma. But we have fun found on the registration study and also now in real world practice experience is that we do not see a decreased response to carty in older patients compared to younger, which is great and we do not necessarily see a worse safety signal. I think that's because we look more at the overall health status or the fitness of the patients. Then we have immunotherapy with by specific where the antibody attaches to the plasma cell on the one side and also to your immune system t cells on the other side bringing the two together so that your own immune system can then fight you're my alone. And this by specific antibodies are being developed by a number of different drug companies and mayo is involved in many of these clinical trials. By specific antibodies. Maybe something that's more accessible to other oncology practices. You know, it's an off the shelf drug that can be infused. So what is a role for that when it's the right time to use that and sequence that with all the treatment options. So we get the biggest benefit for the patient to our group has contributed hugely to our understanding of multiple myeloma and monochrome gamma. These understanding the prevalence of myeloma and related disorders, the risk of progression of pre malignant plasma cell disorders. The new diagnostic criteria for each of these disorders, various treatments that are used currently for the treatment of myeloma and our role in developing them treatments are improving pretty much every year or two. So that whatever prognosis we tell patients today is already outdated because new treatments could change or improve the outcome even next year. Our hope is to prolong survival as much as we can while keeping our eyes focused on the ultimate vision, which is to cure my alone.
S. Vincent Rajkumar, MD
Hematologist
Yi Lin, MD, PhD
Cholangiocarcinoma discoveries and treatment approaches
Rory L. Smoot, M.D., Mayo Clinic surgical oncologist, discusses the discoveries made in the treatment of cholangiocarcinoma and other biliary tract cancers.
Trends and treatment options in colorectal liver metastases
Mayo Clinic surgical oncologist Sean P. Cleary, M.D., discusses the safe and aggressive treatment options available when colorectal cancer has metastasized to the liver.
Head and neck cancer: De-escalation and transoral surgery
Mayo Clinic head and neck cancer specialists discuss de-escalation and transoral surgery treatment advances and the benefits to patients.
Head and neck 3D printing for reconstruction
Mayo Clinic specialists share their approach to head and neck cancer treatment and reconstruction using 3D printing.
Innovative treatments and risk adaptive therapies for myeloma
Mayo Clinic myeloma specialists discuss the latest in treatment and risk adaptive therapies.
Cutting edge approaches to lymphoma care
Mayo Clinic specialists share the latest in lymphoma treatment including chimeric antigen receptor (CAR)-T cell therapy.
Collaborating in care
Collaborate to cure, connect and transform the diagnosis and treatment of serious, complex and rare conditions.
Use of radiotherapies in breast cancer
Learn about radiotherapies in breast cancer.
Strategies for effective breast cancer treatment
Learn about the latest strategies for effective breast cancer treatment.
Innovations in surgical protocols for breast cancer
Learn about innovations in surgical protocols for breast cancer.
Exploring clinical trials for breast cancer
Learn about the latest clinical trials for breast cancer.
Signs of high risk breast cancer
Learn about the latest innovations for high risk breast cancer.
Symptoms and treatment of pancreatic neuroendocrine tumors
How to spot the symptoms of pancreatic neuroendocrine tumors so your patients can receive the right treatment.
Treatment innovations for pancreatic adenocarcinoma
Learn about the latest treatment innovations for pancreatic adenocarcinoma and how these innovations have extended survival.


text:
Steve Maclin Takes Pride In His Work, Wants Fans To Realize What He Doesn't Isn't 'Fake'
Steve Maclin Takes Pride In His Work, Wants Fans To Realize What He Doesn't Isn't 'Fake'
Steve Maclin is all about keeping up appearances and making the most out of his professional wrestling career.
WrestleZone spoke with Steve Maclin to promote IMPACT Wrestling's recent television tapings in Orlando. Maclin is considered to be a heel on air, but he's far from a bad guy when the cameras are off. Asked if he'd consider portraying a babyface and the dynamic of playing the opposite, Maclin said he's very invested in making sure fans get everything they hoped for at shows.
"Once I go through that curtain, it's go time. That's work. Even when I'm behind the scenes or doing signings, it's still work time, but the fans are paying to see me to get an autograph and meet me. There I am interacting with you, so you're paying to see me with the time you're allotted, and I try to make the most of it," Maclin said. "I wanna make the most when I meet fans behind the scenes and I'm gonna be nice to you because I was that fan at one time. Where I was going up to shows early at the Continental Airlines Arena, which became the Izod Center in East Rutherford, and I would try and show up and see the wrestlers come in early. I love wrestling, I love being a fan myself. But when I am out there, it's go time, and I have one goal and that is to be the IMPACT World Champion.
"I want to suspend the disbelief and make everything that I do mean something. When they leave seeing me, I want them to realize that it's not the 'fake' word," Maclin added. "I want people to come across like, 'Okay, he just kicked the shit out of him.' So it's one of those things that I take pride in doing my job. But if I was to be a babyface, nothing would probably change. Maybe I'd smile a little more, I don't know."
Steve Maclin previously told SEScoops' Ella Jay that he was pursuing his bachelor's degree in psychology and gave an update on how that goal is going.
"It's going, man, it's just—school work and me is not the best [relationship]. I'm a very procrastinating person when it comes to work; I like to get it all done last minute. But it's going well. I enjoy it. The one thing I like about online school too is there's more of an open discussion, and being able to talk like this, even though we're typing through words and stuff. But it's fun, and it's another thing to add to later on in life. It's nothing to fall back on it, but it's something to have down the road if I was to pursue something outside of wrestling. It's also let me get more in-tuned with what's going on in the world with mental health and understanding, especially being from military, and a lot of my buddies with PTSD, myself included, dealing with with troubles, getting out of the Marine Corp. and transitioning to real life. Trying to figure out where you fit in, what's this, what's that. What's the reason for this? My mind is really on that outside of work. Other than that, it lets me communicate a lot better with fans."
Asked if he'd ever consider balancing a career in psychology and as a wrestler, Maclin ruled that out and said he's very much focused on what he's doing in the ring now.
"I would say after [I'm done wrestling] just because I have to have everything in one pot right now. I have every bit of me in one hundred percent of wrestling. When I got let go from WWE, I was looking for a 9-5 job to keep paying the bills, trying to figure something out," Maclin said. "Luckily, my time in IMPACT, I got there right away. Within a day, I had my contract there. Everything is in wrestling right now."
Read More: Women's Wrestling Wrap-Up: Royal Rumble Preview, Saraya & Toni Storm Turn Heel, Kaia McKenna Interview
The post Steve Maclin Takes Pride In His Work, Wants Fans To Realize What He Doesn't Isn't 'Fake' appeared first on Wrestlezone.
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text:
package cn.ucloud.ufile.body;

public class PartBody {
    private int PartNumber;

    public PartBody() {

    }

    public int getPartNumber() {
        return PartNumber;
    }

    public void setPartNumber(int partNumber) {
        PartNumber = partNumber;
    }
}

redpajama_set_name: RedPajamaGithub


New Delhi, Apr 4: The compounded annual growth rate for the client PC market from 2006-10 for Pakistan, Bangladesh, and Sri Lanka will reach 19 per cent, 22 per cent and 11 per cent respectively, according to market intelligence and advisory services company, IDC.
''This will be largely driven by public sector purchases as well as the telecommunications and financial services sectors.
The growth rates for 2005 were 19 per cent, 13 per cent, and 12 per cent respectively, IDC said.
Despite some dampeners in 2005, such as risky peace talks and the after-effects of the tsunami, the PC market in Sri Lanka exhibited resiliency largely due to public sector and non-government organisation purchases,'' said IDC's Andrew Wong.
Client PC market in Sri Lanka totaled 393,184 units in the second half of 2005, thus bringing full year 2005 shipments to 851,735 units, representing a 16 per cent growth for the year.
''In Pakistan, encouraging government-led policies and structural liberalisation in the financial and telecommunications sectors helped to lift the PC market there. Bangladesh, on the other hand, was relatively unaffected by the influx of second-hand PCs, thus allowing the market to move ahead,'' he said.
With full year growth of 19 per cent, Pakistan boasted the highest growth rate among the three markets during 2005.
The CAGRs expected in these three countries make the rest of the APEJ market look dull in comparison. ''Pakistan's total market size could exceed the Singapore PC market in absolute terms as early as 2007,'' said IDC's Bryan Ma.
In Pakistan, local branded PC vendors such as In-Box and Raffles grew in 2005 by banking on government and education projects. Multinational PC vendor HP grew 38.5 per cent year-on-year in 2005 due to projects in the government, corporate, and education sectors.
Similarly, HP overtook local player Daffodil in Bangladesh in 2005, where the government, education, finance, and agriculture sectors generated multiple small-scale projects throughout the year, IDC commented.