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arXiv:1902.02602 · Physical Review A

Efficient Bayesian credible-region certification for quantum-state tomography

Changhun Oh, Yong Siah Teo, Hyunseok Jeong

Published Jul 26, 2019• 10 citations
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Abstract

Standard Bayesian credible-region theory for constructing an error region on the unique estimator of an unknown state in general quantum-state tomography to calculate its size and credibility relies on heavy Monte Carlo sampling of the state space followed by filtering to obtain the correct region sample. This conventional methodology typically gives negligible yield for very small error regions originating from large data sets. In this article, we discuss at length the in-region sampling theory for computing both size and credibility from region-average quantities that avoids this general problem altogether. Among the many possible numerical choices, we study the performance and properties of accelerated hit-and-run Monte Carlo algorithm for in-region sampling and provide its complexity estimates for quantum states. Finally with our in-region concept, by alternatively quantifying the region capacity with the region-average distance between two states in the region (measured for instance with either the Hilbert-Schmidt, trace-class, or Bures distance), we derive approximation formulas to analytically estimate both region capacity and credibility without Monte Carlo computation.

DOI
10.1103/physreva.100.012345

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Standard Bayesian credible-region theory for constructing an error region on the unique estimator of an unknown state in general quantum-state tomography to calculate its size and credibility relies on heavy Monte Carlo sampling of the state space followed by filtering to obtain the correct region sample. This conventional methodology typically gives negligible yield for very small error regions originating from large data sets. In this article, we discuss at length the in-region sampling theory for computing both size and credibility from region-average quantities that avoids this general problem altogether. Among the many possible numerical choices, we study the performance and properties of accelerated hit-and-run Monte Carlo algorithm for in-region sampling and provide its complexity estimates for quantum states. Finally with our in-region concept, by alternatively quantifying the region capacity with the region-average distance between two states in the region (measured for instance with either the Hilbert-Schmidt, trace-class, or Bures distance), we derive approximation formulas to analytically estimate both region capacity and credibility without Monte Carlo computation.
Standard computation of size and credibility of a Bayesian credible region for certifying any point estimator of an unknown parameter (such as a quantum state, channel, phase, etc.) requires selecting points that are in the region from a finite parameter-space sample, which is infeasible for a large dataset or dimension as the region would then be extremely small. We solve this problem by introducing the in-region sampling theory to compute both region qualities just by sampling appropriate functions over the region itself using any Monte Carlo sampling method. We take in-region sampling to the next level by understanding the credible-region capacity (an alternative description for the region content to size) as the average l_{p}-norm distance (p>0) between a random region point and the estimator, and present analytical formulas for p=2 to estimate both the capacity and credibility for any dimension and a sufficiently large dataset without Monte Carlo sampling, thereby providing a quick alternative to Bayesian certification. All results are discussed in the context of quantum-state tomography.
Results concerning the construction of quantum Bayesian error regions as a means to certify the quality of parameter point estimators have been reported in recent years. This task remains numerically formidable in practice for large dimensions and so far, no analytical expressions of the region size and credibility (probability of any given true parameter residing in the region) are known, which form the two principal region properties to be reported alongside a point estimator obtained from collected data. We first establish analytical formulas for the size and credibility that are valid for a uniform prior distribution over parameters, sufficiently large data samples and general constrained convex parameter estimation settings. These formulas provide a means to an efficient asymptotic error certification for parameters of arbitrary dimensions. Next, we demonstrate the accuracies of these analytical formulas as compared to numerically computed region quantities with simulated examples in qubit and qutrit quantum-state tomography where computations of the latter are feasible.
An estimator is a state that represents one's best guess of the actual state of the quantum system for the given data. Such estimators are points in the state space. To be statistically meaningful, they have to be endowed with error regions, the generalization of error bars beyond one dimension. As opposed to standard ad hoc constructions of error regions, we introduce the maximum-likelihood region—the region of largest likelihood among all regions of the same size—as the natural counterpart of the popular maximum-likelihood estimator. Here, the size of a region is its prior probability. A related concept is the smallest credible region—the smallest region with pre-chosen posterior probability. In both cases, the optimal error region has constant likelihood on its boundary. This surprisingly simple characterization permits concise reporting of the error regions, even in high-dimensional problems. For illustration, we identify optimal error regions for single-qubit and two-qubit states from computer-generated data that simulate incomplete tomography with few measured copies.
Bayesian error analysis paves the way to the construction of credible and plausible error regions for a point estimator obtained from a given dataset. We introduce the concept of region accuracy for error regions (a generalization of the point-estimator mean squared error) to quantify the average statistical accuracy of all region points with respect to the unknown true parameter. We show that the increase in region accuracy is closely related to the Bayesian region dual operations in Shang et al (2013 New J. Phys. 15 123026). Next with only the given dataset as viable evidence, we establish various adaptive methods to maximize the region accuracy relative to the true parameter subject to the type of reported Bayesian region for a given point estimator. We highlight the performance of these adaptive methods by comparing them with non-adaptive procedures in three quantum-parameter estimation examples. The results of and mechanisms behind the adaptive schemes can be understood as the region analog of adaptive approaches to achieving the quantum Cramér–Rao bound for point estimators.
Quantum state tomography is the standard technique for reconstructing a quantum state from experimental data. In the regime of finite statistics, experimental data cannot give perfect information about the quantum state. A common way to express this limited knowledge is by providing confidence regions in the state space. Though other confidence regions were previously proposed, they are either too wasteful to be of practical interest, cannot easily be applied to general measurement schemes, or are too difficult to report. Here we construct confidence regions that solve these issues, as they have an asymptotically optimal sample cost and good performance for realistic parameters, are applicable to any measurement scheme, and can be described by an ellipsoid in the space of Hermitian operators. Our construction relies on a vector Bernstein inequality and bounds with high probability the Hilbert--Schmidt norm error of sums of multinomial samples transformed by linear maps.
The outcomes of quantum mechanical measurements are inherently random. It is therefore necessary to develop stringent methods for quantifying the degree of statistical uncertainty about the results of quantum experiments. For the particularly relevant task of quantum state tomography, it has been shown that a significant reduction in uncertainty can be achieved by taking the positivity of quantum states into account. However—the large number of partial results and heuristics notwithstanding—no efficient general algorithm is known that produces an optimal uncertainty region from experimental data, while making use of the prior constraint of positivity. Here, we provide a precise formulation of this problem and show that the general case is NP-hard. Our result leaves room for the existence of efficient approximate solutions, and therefore does not in itself imply that the practical task of quantum uncertainty quantification is intractable. However, it does show that there exists a non-trivial trade-off between optimality and computational efficiency for error regions. We prove two versions of the result: one for frequentist and one for Bayesian statistics.