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Minimax Optimal Estimation of Expectation Values

Seshadri, Akshay

Published Apr 15, 2025• 0 citations
OpenAlex

Abstract

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.

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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.
Estimating the fidelity with a target state is important in quantum information tasks. Many fidelity-estimation techniques present a suitable measurement scheme to perform the estimation. In contrast, we present techniques that allow the experimentalist to choose a convenient measurement setting. Our primary focus lies on a method that constructs an estimator with nearly minimax optimal confidence intervals for any specified measurement setting. We demonstrate, through a combination of theoretical and numerical results, various desirable properties of the method: robustness against experimental imperfections, competitive sample complexity, and accurate estimates in practice. We compare this method with maximum likelihood estimation and the associated profile likelihood method, a semi-definite programming based approach, direct fidelity estimation, quantum state verification, and classical shadows. Our method can also be used for estimating the expectation value of any observable with the same guarantees.
As quantum devices become more complex and the requirements on these devices become more demanding, it is crucial to be able to verify the performance of such devices in a scalable and reliable fashion. A cornerstone task in this challenge is quantifying how close an experimentally prepared quantum state is to the desired one. Here we present a method to construct an estimator for the quantum state fidelity that is compatible with any measurement protocol. Our method provides a confidence interval on this estimator that is guaranteed to be nearly minimax optimal for the specified measurement protocol. For a well-chosen measurement scheme, our method is competitive in the number of measurement outcomes required for estimation. We demonstrate our method using simulations and experimental data from a trapped-ion quantum computer and compare the results to state-of-the-art techniques. Our method can be easily extended to estimate the expectation value of any observable, such as entanglement witnesses.
Improving the performance of quantum algorithms is a fundamental task to achieve quantum advantage. In many cases, extracting information from quantum systems poses an important challenge for practical implementations of real-world quantum computers, given the high resource cost of performing state tomography. In this scenario, randomized measurements have emerged as a promising tool. In particular, the classical shadow protocol allows one to retrieve expected values of low-weight Pauli observables by performing only local measurements. In this paper, we present an implementation of the recently introduced feedback-based algorithm for quantum optimization using the Ket quantum programming platform to solve the MaxCut optimization problem. We employ classical shadows for the feedback routine of parameter estimation and compare this approach with the direct estimation of observables. Our results show that depending on the graph geometry of the MaxCut problem, the number of measurements required to estimate expected values of observables with classical shadows can be up to 16 times lower than with direct-observable estimation. Furthermore, by analyzing complete graphs, we numerically confirm logarithmic growth in the required number of measurements relative to the number of observables, reinforcing that classical shadows can be a useful tool for estimating low-locality Pauli observables in quantum algorithms.
Estimation of the expectation value of observables is a key subroutine in quantum computing and is also the bottleneck of the performance of many near-term quantum algorithms. Many methods have been proposed to reduce the number of measurements needed for this task by designing measurement schemes that decide the measurements to perform; however, these schemes are usually constructed from hand-crafted heuristics, which limits the measurement efficiency they can achieve. In this paper, we propose a framework for learning measurement schemes directly from the observable, using machine learning techniques including stochastic gradient descent and a two time-scale update rule. As a concrete realization of this framework, we introduce Composite-Locally Biased Classical Shadow (C-LBCS), which learns a mixture of locally-biased classical shadows and their mixing weights end-to-end. We numerically demonstrate C-LBCS on molecular systems up to CO2\mathrm{CO}_2CO2​ (30 qubits) and show that C-LBCS outperforms the previous state-of-the-art methods despite its simplicity. We believe our approach opens up a reliable and scalable path toward efficient observable estimation on large quantum systems.
Efficiently estimating properties of large and strongly coupled quantum systems is a central focus in many-body physics and quantum information theory.While quantum computers promise speedups for many of these tasks, near-term devices are prone to noise that will generally reduce the accuracy of such estimates.Here, we propose a sample-efficient and noise-resilient protocol for learning properties of quantum states building on the shadow estimation scheme [Huang et al., Nature Physics 16, 1050-1057(2020)].By introducing an experimentally friendly calibration procedure, our protocol can efficiently characterize and mitigate noises in the shadow estimation scheme, given only minimal assumptions on the experimental conditions.When the strength of noises can be bounded, our protocol approximately retains the same order of sample efficiency as the standard shadow estimation scheme, while also possessing a provable noise resilience.We give rigorous bounds on the sample complexity of our protocol and demonstrate its performance with several numerical experiments, including estimations of quantum fidelity, correlation functions and energy expectations, etc., which highlight a wide spectrum of potential applications of our protocol on near-term devices.
Efficiently estimating properties of large and strongly coupled quantum systems is a central focus in many-body physics and quantum information theory. While quantum computers promise speedups for many of these tasks, near-term devices are prone to noise that will generally reduce the accuracy of such estimates. Here, we propose a sample-efficient and noise-resilient protocol for learning properties of quantum states building on the shadow estimation scheme [Huang et al., Nature Physics 16, 1050-1057 (2020)]. By introducing an experimentally friendly calibration procedure, our protocol can efficiently characterize and mitigate noises in the shadow estimation scheme, given only minimal assumptions on the experimental conditions. When the strength of noises can be bounded, our protocol approximately retains the same order of sample efficiency as the standard shadow estimation scheme, while also possessing a provable noise resilience. We give rigorous bounds on the sample complexity of our protocol and demonstrate its performance with several numerical experiments, including estimations of quantum fidelity, correlation functions and energy expectations, etc., which highlight a wide spectrum of potential applications of our protocol on near-term devices.