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.
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.
Direct fidelity estimation provides a way to estimate the fidelity between an experimentally prepared state and a desired pure target state without performing full tomography. Two influential formulations were introduced in 2011 by Flammia and Liu and by da Silva, Landon-Cardinal, and Poulin. In these protocols, the total estimation error is controlled through two distinct probabilistic steps: first, the fidelity is approximated using randomly sampled Pauli observables; second, each sampled expectation value is estimated from finitely many measurement outcomes. In this work we show that additional structural information about the noise can substantially sharpen the corresponding resource bounds. In particular, for some canonical channels the effective number of sampled Pauli settings can be reduced, leading to lower measurement cost both in the general pure-state setting and in the case of a stabilizer state. These results illustrate a broader point: worst-case confidence bounds in direct fidelity estimation can be significantly conservative when experimentally relevant structure is ignored. As a technical ingredient, we also revisit the allocation of the total accuracy and confidence budgets between the two probabilistic steps. Reformulating the analysis in terms of separate error parameters yields a constrained optimization problem whose solution lowers the average number of measurements in the general pure-state setting. Numerical simulations based on quantum circuits implemented in Qiskit illustrate both the improvement obtained under structured-noise assumptions and the conservativeness of the original worst-case bounds.
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.
Quantum fidelity estimation is a fundamental task in quantum information and computing to benchmark the quality of quantum states, and hence of quantum processes, on noisy quantum hardware. Point estimates of fidelity by existing methods generally lack the rigorous uncertainty quantification required for reliable preparation and verification of these tasks. While Bayesian and bootstrap frameworks are widely used for uncertainty quantification, they are sensitive to model misspecification or lack formal coverage guarantees in non-asymptotic regimes. This paper introduces a hybrid "Bayesian-informed" conformal prediction-based framework to provide distribution-free, rigorous prediction intervals for quantum state fidelity estimation which can be used, with care, to approximate confidence intervals with respect to the quantum randomness of Born's rule. The framework requires a point estimator, and we consider both classical point estimators as well as data-driven neural net estimators. We evaluate the robustness of the methodology across a diverse range of experimental challenges, including finite-sample noise, tomographic incompleteness, physical model error, and out-of-distribution data. Our results demonstrate that conformalized estimators maintain statistical validity where traditional methods fail, offering a scalable and reliable approach for the verification of modern quantum systems. The best performing variant of our methodology is based on a class-conditional conformal prediction, as this most closely approximates the conditional coverage desired for a true confidence interval.
Reliable characterization of quantum states and quantum processes is central to the development of scalable quantum computing. Standard characterization techniques such as full quantum state tomography become rapidly impractical as system size increases. In this work we present two software-oriented fidelity estimation protocols that avoid full tomographic reconstruction while remaining experimentally relevant for noisy intermediate-scale quantum devices. The first protocol is direct fidelity estimation (DFE), where the fidelity between a target state and an experimentally prepared state is estimated from importance-sampled Pauli measurements. The second protocol is a survival-probability-based method that reconstructs fidelity from experimentally accessible moments of a noisy evolution and its approximate inverse, and is intrinsically robust against state-preparation-and-measurement (SPAM) errors. We discuss the theoretical basis of both approaches, analyze their noise sensitivity, and present numerical evidence on representative multiqubit states under depolarizing, dephasing, and amplitude-damping channels. For DFE we show that attenuation-aware correction improves robustness against measurement noise. For the survival-probability approach we show that an L2-optimized expansion substantially outperforms naive Taylor truncation in high-noise regimes.
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.