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arXiv:2304.10949 · Journal of Physics A Mathematical and Theoretical

Quantum information criteria for model selection in quantum state estimation

Hiroshi Yano, Naoki Yamamoto

Published Sep 6, 2023• 4 citations
arXivHTMLPDF

Abstract

Abstract Quantum state estimation (or state tomography) is an indispensable task in quantum information processing. Because full state tomography that determines all elements of the density matrix is computationally demanding, one usually takes the strategy of assuming a certain model of quantum states and identifying the model parameters. However, it is difficult to make a valid assumption given little prior knowledge on a quantum state of interest, and thus we need a reasonable model selection method for quantum state estimation. Actually, in the classical statistical estimation theory, several types of information criteria have been established and widely used in practice for appropriately choosing a classical statistical model. In this study, we propose quantum information criteria for evaluating the quality of the estimated quantum state in terms of the quantum relative entropy, which is a natural quantum analogue of the classical information criterion defined in terms of Kullback–Leibler divergence. In particular, we derive two quantum information criteria depending on the type of an estimator for the quantum relative entropy; one uses the log-likelihood and the other uses the classical shadow. The general role of information criteria is to predict the performance of an estimated model for unseen data, although it is a function of only sampled data; this generalization capability of the proposed quantum information criteria is evaluated in numerical simulations.

DOI
10.1088/1751-8121/acf747

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Abstract Quantum state estimation (or state tomography) is an indispensable task in quantum information processing. Because full state tomography that determines all elements of the density matrix is computationally demanding, one usually takes the strategy of assuming a certain model of quantum states and identifying the model parameters. However, it is difficult to make a valid assumption given little prior knowledge on a quantum state of interest, and thus we need a reasonable model selection method for quantum state estimation. Actually, in the classical statistical estimation theory, several types of information criteria have been established and widely used in practice for appropriately choosing a classical statistical model. In this study, we propose quantum information criteria for evaluating the quality of the estimated quantum state in terms of the quantum relative entropy, which is a natural quantum analogue of the classical information criterion defined in terms of Kullback–Leibler divergence. In particular, we derive two quantum information criteria depending on the type of an estimator for the quantum relative entropy; one uses the log-likelihood and the other uses the classical shadow. The general role of information criteria is to predict the performance of an estimated model for unseen data, although it is a function of only sampled data; this generalization capability of the proposed quantum information criteria is evaluated in numerical simulations.
Recently several more efficient versions of quantum state tomography have been proposed, with the purpose of making tomography feasible even for many-qubit states. The number of state parameters to be estimated is reduced by tentatively introducing certain simplifying assumptions on the form of the quantum state, and subsequently using the data to rigorously verify these assumptions. The simplifying assumptions considered so far were (i) the state can be well approximated to be of low rank, or (ii) the state can be well approximated as a matrix product state, or (iii) only the permutationally invariant part of the density matrix is determined. We add one more method in that same spirit: We allow in principle any model for the state, using any (small) number of parameters (which can, e.g., be chosen to have a clear physical meaning), and the data are used to verify the model. The proof that this method is valid cannot be as strict as in the above-mentioned cases, but is based on well-established statistical methods that go under the name of ``information criteria.'' We exploit here, in particular, the Akaike information criterion. We illustrate the method by simulating experiments on (noisy) Dicke states.
Deep learning has seen substantial achievements, with numerical and theoretical evidence suggesting that singularities of statistical models are considered a contributing factor to its performance. From this remarkable success of classical statistical models, it is naturally expected that quantum singular models will play a vital role in many quantum statistical tasks. However, while the theory of quantum statistical models in regular cases has been established, theoretical understanding of quantum singular models is still limited. To investigate the statistical properties of quantum singular models, we focus on two prominent tasks in quantum statistical inference: quantum state estimation and model selection. In particular, we base our study on classical singular learning theory and seek to extend it within the framework of Bayesian quantum state estimation. To this end, we define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods. The key idea of the proof is the introduction of a quantum analog of the likelihood function using classical shadows. Consequently, we construct an asymptotically unbiased estimator of the quantum generalization loss, the quantum widely applicable information criterion (QWAIC), as a computable model selection metric from given measurement outcomes.
Quantum state estimation has been widely investigated and there are mainly two approaches proposed: One is based on the point estimation of an unknown parameter and the other is based on the Bayesian method. We adopt the relative entropy from the true state to a predictive density operator as a loss function. We consider exchangeable quantum models with an arbitrary chosen measurement and show that Bayesian predictive density operators are the best predictive density operators when we evaluate them by using the average relative entropy based on a prior. This result is a quantum version of Aitchison's result in classical statistics.
Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state and, therefore, have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in terms of confidence regions-a concept originating from classical statistics. Confidence regions are subsets of the state space in which the true state lies with high probability, independently of any prior assumption on the distribution of the possible states. Our method for computing confidence regions can be applied to arbitrary measurements including fully coherent ones; it is practical and particularly well suited for tomography on systems consisting of a small number of qubits, which are currently in the focus of interest in experimental quantum information science.
In the quantum state tomography, which is the task of estimating the unknown quantum state on a Hilbert space from measurement data, the standard Bayesian technique is often used.As is well known in Bayesian statistics, it is a nontrivial problem to define a noninformative prior in gen- eral statistical models.Although some definitions for statistical models of mixed states are given by straightforward extension of classical results, there is no counterpart for those of pure states.Recently the author proposed a conceptual extension of the famous reference prior by Bemardo and defined noninformative priors for statistical models of pure states.While the formal extension of the reference prior corresponds to the prior maximizing the von Neumann entropy, our proposed prior is shown to maximize the limit of the quantum Renyi entropy.We show some relations between two principles.
Quantum computation has been growing rapidly in both theory and experiments.In particular, quantum computing devices with a large number of qubits have been developed by IBM, Google, IonQ, and others.The current quantum computing devices are noisy intermediate-scale quantum devices, and so approaches to validate quantum processing on these quantum devices are needed.One of the most common ways of validation for an n-qubit quantum system is quantum tomography, which tries to reconstruct a quantum system's density matrix by a complete set of observables.However, the inherent noise in the quantum systems and the intrinsic limitations pose a critical challenge to precisely know the actual measurement operators that make quantum tomography impractical in experiments.Here, we propose an alternative approach to quantum tomography, based on the maximal information entropy, that can predict the values of unknown observables based on the available mean measurement data.This can then be used to reconstruct the density matrix with high fidelity even though the results for some observables are missing.Of additional contexts, a practical approach to the inference of the quantum mechanical state using only partial information is also needed.