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arXiv:1705.03853 · Physical Review A

Exponential families for Bayesian quantum process tomography

Kevin Schultz

Published Dec 11, 2019• 9 citations
arXivHTMLPDF

Abstract

A Bayesian approach to quantum process tomography has yet to be fully developed due to the lack of appropriate probability distributions on the space of quantum channels. Here, by associating the Choi matrix form of a completely positive, trace preserving (CPTP) map with a particular space of matrices with orthonormal columns, called a Stiefel manifold, we present two parametric probability distributions on the space of CPTP maps that enable Bayesian analysis of process tomography. The first is a probability distribution that has an average Choi matrix as a sufficient statistic. The second is a distribution resulting from binomial likelihood data that enables a simple connection to data gathered through process tomography experiments. These are examples of continuous, nonunitary random CPTP maps that capture meaningful prior information for use in Bayesian estimation. We show how these distributions can be used for point estimation using either maximum a posteriori estimates or expected a posteriori estimates, as well as full Bayesian tomography resulting in posterior credibility intervals. This approach will enable the full power of Bayesian analysis in all forms of quantum characterization, verification, and validation.

DOI
10.1103/physreva.100.062316

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A Bayesian approach to quantum process tomography has yet to be fully developed due to the lack of appropriate probability distributions on the space of quantum channels. Here, by associating the Choi matrix form of a completely positive, trace preserving (CPTP) map with a particular space of matrices with orthonormal columns, called a Stiefel manifold, we present two parametric probability distributions on the space of CPTP maps that enable Bayesian analysis of process tomography. The first is a probability distribution that has an average Choi matrix as a sufficient statistic. The second is a distribution resulting from binomial likelihood data that enables a simple connection to data gathered through process tomography experiments. These are examples of continuous, nonunitary random CPTP maps that capture meaningful prior information for use in Bayesian estimation. We show how these distributions can be used for point estimation using either maximum a posteriori estimates or expected a posteriori estimates, as well as full Bayesian tomography resulting in posterior credibility intervals. This approach will enable the full power of Bayesian analysis in all forms of quantum characterization, verification, and validation.
Using a Bayesian methodology, we introduce the maximum a posteriori probability (MAP) estimator for quantum state and process tomography. We show that the maximum likelihood, the hedged maximum likelihood, and the maximum likelihood-maximum entropy estimator, and estimators of this general type, can be viewed as special cases of the MAP estimator. The MAP, like the Bayes mean estimator, includes prior knowledge. For cases of interest to tomography MAP can take advantage of convex optimization tools, making it numerically tractable. We show how the MAP and other Bayesian quantum state estimators can be corrected for noise produced if the quantum state passes through a noisy quantum channel prior to measurement. Numerical simulations on a single qubit indicate that, on average, including these corrections significantly improves the estimate even when the measurement data are modestly large.
Abstract The Choi representation of completely positive trace preserving (CPTP) maps, i.e. quantum channels is often used in the context of quantum information and computation as it is easy to work with. It is a correspondence between CPTP maps and quantum states also termed as the Choi–Jamiołkowski isomorphism. It is especially useful if a parametrization of the set of CPTP maps is needed in order to consider a general map or optimize over the set of these. Here we provide a brief introduction to this topic, focusing on certain useful calculational techniques which are presented in full detail.
In recent years, Bayesian methods have been proposed as a solution to a wide range of issues in quantum state and process tomography. State-of-the-art Bayesian tomography solutions suffer from three problems: numerical intractability, a lack of informative prior distributions, and an inability to track time-dependent processes. Here, we address all three problems. First, we use modern statistical methods, as pioneered by Huszár and Houlsby (2012 Phys. Rev. A 85 052120 ) and by Ferrie (2014 New J. Phys. 16 093035 ), to make Bayesian tomography numerically tractable. Our approach allows for practical computation of Bayesian point and region estimators for quantum states and channels. Second, we propose the first priors on quantum states and channels that allow for including useful experimental insight. Finally, we develop a method that allows tracking of time-dependent states and estimates the drift and diffusion processes affecting a state. We provide source code and animated visual examples for our methods.
We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a ddd-level system is completely characterized by a minimal set of d2+d{d}^{2}+dd2+d elements associated with a collection of POVMs, in contrast to the ${d}^{4}\ensuremath{-}{d}^{2}$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with particular sets of ddd pure states. We further compare the performance of different estimators inspired by compressed sensing, to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Convex optimization inspired by the techniques of compressed sensing can therefore be used both as indicators of error models and to validate the use of the prior assumptions.
We perform quantum process tomography (QPT) for both discrete- and continuous-variable quantum systems by learning a process representation using Kraus operators. The Kraus form ensures that the reconstructed process is completely positive. To make the process trace preserving, we use a constrained gradient-descent (GD) approach on the so-called Stiefel manifold during optimization to obtain the Kraus operators. Our ansatz uses a few Kraus operators to avoid direct estimation of large process matrices, e.g., the Choi matrix, for low-rank quantum processes. The GD-QPT matches the performance of both compressed-sensing (CS) and projected least-squares (PLS) QPT in benchmarks with two-qubit random processes, but shines by combining the best features of these two methods. Similar to CS (but unlike PLS), GD-QPT can reconstruct a process from just a small number of random measurements, and similar to PLS (but unlike CS) it also works for larger system sizes, up to at least five qubits. We envisage that the data-driven approach of GD-QPT can become a practical tool that greatly reduces the cost and computational effort for QPT in intermediate-scale quantum systems.
We present a Bayesian approach to the problem of estimating density matrices in quantum state tomography. A general framework is presented based on a suitable mathematical formulation, where a study of the convergence of the Monte Carlo Markov Chain algorithm is given, including a comparison with other estimation methods, such as maximum likelihood estimation and linear inversion. This analysis indicates that our approach not only recovers the underlying parameters quite properly, but also produces physically acceptable punctual and interval estimates. A prior sensitive study was conducted indicating that when useful prior information is available and incorporated, more accurate results are obtained. This general framework, which is based on a reparameterization of the model, allows an easier choice of the prior and proposal distributions for the Metropolis–Hastings algorithm.