Quantum state estimation aims at determining the quantum state from observed data. Estimating the full state can require considerable efforts, but one is often only interested in a few properties of the state, such as the fidelity with a target state, or the degree of correlation for a specified bipartite structure. Rather than first estimating the state, one can, and should, estimate those quantities of interest directly from the data. We propose the use of optimal error intervals as a meaningful way of stating the accuracy of the estimated property values. Optimal error intervals are analogs of the optimal error regions for state estimation [New J. Phys. 15, 123026 (2013)]. They are optimal in two ways: They have the largest likelihood for the observed data and the prechosen size, and they are the smallest for the prechosen probability of containing the true value. As in the state situation, such optimal error intervals admit a simple description in terms of the marginal likelihood for the data for the properties of interest. Here, we present the concept and construction of optimal error intervals, report on an iterative algorithm for reliable computation of the marginal likelihood (a quantity difficult to calculate reliably), explain how plausible intervals---a notion of evidence provided by the data---are related to our optimal error intervals, and illustrate our methods with single-qubit and two-qubit examples.
Quantum state estimation aims at determining the quantum state from observed data. Estimating the full state can require considerable efforts, but one is often only interested in a few properties of the state, such as the fidelity with a target state, or the degree of correlation for a specified bipartite structure. Rather than first estimating the state, one can, and should, estimate those quantities of interest directly from the data. We propose the use of optimal error intervals as a meaningful way of stating the accuracy of the estimated property values. Optimal error intervals are analogs of the optimal error regions for state estimation [New J. Phys. 15, 123026 (2013)]. They are optimal in two ways: They have the largest likelihood for the observed data and the prechosen size, and they are the smallest for the prechosen probability of containing the true value. As in the state situation, such optimal error intervals admit a simple description in terms of the marginal likelihood for the data for the properties of interest. Here, we present the concept and construction of optimal error intervals, report on an iterative algorithm for reliable computation of the marginal likelihood (a quantity difficult to calculate reliably), explain how plausible intervals---a notion of evidence provided by the data---are related to our optimal error intervals, and illustrate our methods with single-qubit and two-qubit examples.
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.
There is growing interest in constructing quantum devices that control increasingly large numbers of quantum systems. There is a tremendous need for methods that can measure the quality and other properties of these devices. Such measurements often amount to learning the expectation value of a quantum observable with respect to the quantum state of the device. Learning expectation values is also a key component in many well-known applications such as quantum machine learning and quantum optimization algorithms. Because of the large system sizes involved, it is essential to find methods for learning expectation values that are efficient with respect to the system size. Existing methods that have rigorous guarantees and are practical to implement are often not optimal for observables of interest. Other methods are heuristic, or require the use measurement protocols that are challenging to implement experimentally with current technology. In this study, we propose an estimation procedure, The Optimal Observable expectation value Learner or TOOL, that can learn the expectation values of observables using the outcomes of any given measurement protocol. We show that there is a seminorm on the set of all observables, which we call the minimax norm, that characterizes the smallest possible estimation error for learning a given observable using the outcomes of a given non-adaptive measurement protocol to a specified confidence level in the worst case over all states. We prove that TOOL is minimax optimal for every observable by showing that it can achieve an estimation error to within a small constant factor of the minimax norm. For many applications, one wishes to learn the expectation value of more than one observable from the same experiment. A popular method for learning the expectation values of one or many observables with rigorous guarantees is classical shadows. Classical shadows has near-optimal performance in the worst case over all observables. We prove that TOOL always performs at least as well as classical shadows. Moreover, we show by example that TOOL dramatically outperforms classical shadows for many observables of interest. This highlights the need to characterize the optimal performance for the task of simultaneously learning the expectation values of many observables. Under a mild assumption, we give such a characterization using the minimax norm and prove that TOOL is nearly minimax optimal for this task. We also study the applications of TOOL to fidelity estimation. Using experimental data from a trapped-ion quantum computer, we show that TOOL performs well in practice and matches the estimates obtained from Maximum Likelihood Estimation (MLE), but with rigorous guarantees on the estimation error unlike MLE. We also compare TOOL with another popular method called direct fidelity estimation, which estimates the fidelity by judiciously sampling Pauli observables and measuring them. We show that there is a different importance sampling scheme for Pauli measurements for which TOOL performs as well as, or better than, direct fidelity estimation. Since TOOL constructs an estimator using only the observable, the measurement protocol, and the confidence level, it provides the flexibility to perform estimation for experiments that have already been performed and experiments that will be performed in the future. Similarly, since the minimax norm can be computed beforehand, it can be used to compare the performance of different measurement protocols and allow minimax optimal design of experiments.
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 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.
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.