Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0607019 · Physical Review A

Optimal quantum measurements of expectation values of observables

Emanuel H. Knill, Gerardo Ortíz, Rolando D. Somma

Published Jan 24, 2007• 195 citations
arXivHTMLPDF

Abstract

Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state $\ensuremath{\mid}\ensuremath{\psi}⟩$ of the quantum system. Such expectation values can be measured by repeatedly preparing $\ensuremath{\mid}\ensuremath{\psi}⟩$ and coupling the system to an apparatus. For this method, the precision of the measured value scales as $\frac{1}{\sqrt{N}}$ for $N$ repetitions of the experiment. For the problem of estimating the parameter $\ensuremath{\phi}$ in an evolution ${e}^{\ensuremath{-}i\ensuremath{\phi}H}$, it is possible to achieve precision $\frac{1}{N}$ [the quantum metrology limit; see Giovannetti et al., Phys. Rev. Lett. 96, 010401 (2006)] provided that sufficient information about $H$ and its spectrum is available. We consider the more general problem of estimating expectations of operators $A$ with minimal prior knowledge of $A$. We give explicit algorithms that approach precision $\frac{1}{N}$ given a bound on the eigenvalues of $A$ or on their tail distribution. These algorithms are particularly useful for simulating quantum systems on quantum computers because they enable efficient measurement of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of $\ensuremath{\mid}\ensuremath{\psi}⟩$ and $U\ensuremath{\mid}\ensuremath{\psi}⟩$, where $U$ is an implementable unitary operator. We explicitly consider the issue of confidence levels in measuring observables and overlaps and show that, as expected, confidence levels can be improved exponentially with linear overhead. We further show that the algorithms given here can typically be parallelized with minimal increase in resource usage.

DOI
10.1103/physreva.75.012328

Related papers

Minimax Optimal Estimation of Expectation Values

Seshadri, Akshay

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.

•Apr 15, 2025•0 citations

Optimal quantum phase estimation algorithms

Joseph T. Smith

It is suggested that computers exploiting quantum phenomena can solve tasks with an efficiency surpassing that achievable by classical computers. One example of this quantum advantage arises in the task of phase estimation. Here, an unknown eigenvalue of a unitary evolution is estimated, a critical subroutine for many quantum algorithms and quantum metrology protocols. For a fixed amount of resources, the utilisation of quantum coherence and entanglement can lead to a quadratic increase in precision of the final estimate compared to classical statistical limits. In systems affected by depolarising noise, including current quantum hardware, this improvement becomes a constant factor. While numerous phase estimation algorithms have been proposed claiming to achieve these theoretical limits, the lack of well-defined criteria for both precision and resource utilisation casts doubt on these results. Further issues arise due to a lack of point identification (the existence of multiple equally likely estimates) that some algorithms have. I aim to address these issues in this thesis. I employ statistical inference, the theoretical foundation for interpreting any quantum measurement, in order to find rigorous definitions for precision and resource use. By combining these definitions with previous work in the field of quantum metrology, I am able to derive theoretical precision limits that one can hope to achieve. I then outline several phase estimation algorithms that can achieve these limits in different contexts. These algorithms that I describe are split into three major groups. Algorithms in the first group focus on the process of point identification; one set of measurements provides a precise estimate of the unknown phase before an additional set of measurements removes any ambiguities present. This second set of measurements can be either decided before any measurement is taken or during measurements in an iterative fashion. I show that the balancing of resources used in each set of measurements can lead to optimally scaling precision both with and without depolarising noise. These algorithms are based upon two of my peer-reviewed articles, [Phys. Rev. A 106, 062615 (2022)] and [Phys. Rev. A 109, 042412 (2024)]. The second group of algorithms that I introduce use the principles of Bayesian experimental design. Here, previous measurement outcomes are used to select future measurements recursively. These algorithms require large classical computing times to select each measurement but boast high precision in certain metrics. The final group of algorithms exploit the oscillating nature of phases by using a discrete Fourier transform (DFT). The DFT can be implemented using either classical or quantum hardware, the latter of which leads to the famous quantum phase estimation algorithm (QPEA). In this thesis, I show that the QPEA performs worse than some classical precision limits. However, I also suggest some modifications to the QPEA circuit that can lead to optimal precision scaling. These results and modifications are described in one of my articles awaiting publication [arXiv:2505.00764].

•Sep 12, 2025•0 citations

An approximate description of quantum states

Marco Paini, Amir Kalev

We introduce an approximate description of an $N$-qubit state, which contains sufficient information to estimate the expectation value of any observable with precision independent of $N$. We show, in fact, that the error in the estimation of the observables' expectation values decreases as the inverse of the square root of the number of the system's identical preparations and increases, at most, linearly in a suitably defined, $N$-independent, seminorm of the observables. Building the approximate description of the $N$-qubit state only requires repetitions of single-qubit rotations followed by single-qubit measurements and can be considered for implementation on today's Noisy Intermediate-Scale Quantum (NISQ) computers. The access to the expectation values of all observables for a given state leads to an efficient variational method for the determination of the minimum eigenvalue of an observable. The method represents one example of the practical significance of the approximate description of the $N$-qubit state. We conclude by briefly discussing extensions to generative modelling and with fermionic operators.

arXiv:1910.10543•Oct 23, 2019•28 citations

Heisenberg-Limited Adaptive Gradient Estimation for Multiple Observables

Kaito Wada, Naoki Yamamoto, Nobuyuki Yoshioka

In quantum mechanics, measuring the expectation value of a general observable has an inherent statistical uncertainty that is quantified by variance or mean squared error of measurement outcome. While the uncertainty can be reduced by averaging several samples, the number of samples should be minimized when each sample is very costly. This is especially the case for fault-tolerant quantum computing that involves measurement of multiple observables of nontrivial states in large quantum systems that exceed the capabilities of classical computers. In this work, we provide an adaptive quantum algorithm for estimating the expectation values of M general observables within root mean squared error ε simultaneously, using O ( ε − 1 M log  M ) queries to a state preparation oracle of a target state. This remarkably achieves the scaling of Heisenberg limit 1 / ε , a fundamental bound on the estimation precision in terms of mean squared error, together with the sublinear scaling of the number of observables M . The proposed method is an adaptive version of the quantum gradient-estimation algorithm and has a resource-efficient implementation due to its adaptiveness. Specifically, the space overhead in the proposed method is O ( M ) , which is independent from the estimation precision ε unlike noniterative algorithms. In addition, our method can avoid the numerical instability problem for constructing quantum circuits in a large-scale task (e.g., ε ≪ 1 in our case), which appears in the actual implementation of many algorithms relying on quantum signal processing techniques. Our method paves a new way to precisely understand and predict various physical properties in complicated quantum systems using quantum computers.

arXiv:2406.03306•PRX Quantum•Apr 10, 2025•3 citations

Efficient expectation value estimation for quantum circuits via extended stabilizer frameworks and adaptive variance estimation

Yunseo Hwang, Giwon Song, Kyoung Keun Park +2 more

Classical simulability of quantum circuits plays a central role in characterizing and quantifying quantum computational advantage. Typically, the presence of non-Clifford gates introduces an exponential runtime overhead for classically estimating expectation values of observables. In this work, we introduce an efficient simulation framework that integrates the sum-over-Clifford method with quasi-probability-based stabilizer simulation to directly estimate expectation values. To optimize efficiency, we incorporate a circuit-adaptive reduction via variance estimation protocol, which terminates simulations once an empirical confidence bound is sufficient to guarantee the prescribed error tolerance. Our method circumvents overly conservative Hoeffding's inequality, thereby reducing the dependence on the gate-wise stabilizer extent from quadratic to linear in the low-variance regime, while maintaining the same space complexity as previous approaches. We explicitly prove the sample-complexity reduction for random quantum circuits and T-doped Clifford circuits and also empirically validate these advantages in the quantum approximate optimization algorithm and quantum kernel method.

quant-pharXiv:2609.14252•Sep 15, 2026•0 citations

Minimal Optimal Generalized Quantum Measurements

José Ignacio Latorre, P. Pascual, R. Tarrach

Optimal and finite positive operator valued measurements on a finite number $N$ of identically prepared systems have recently been presented. With physical realization in mind, we propose here optimal and minimal generalized quantum measurements for two-level systems. We explicitly construct them up to $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}7$ and verify that they are minimal up to $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5$.

arXiv:quant-ph/9803066•Physical Review Letters•Aug 17, 1998•84 citations
Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state \ensuremath∣\ensuremathψ⟩\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath∣\ensuremathψ⟩ of the quantum system. Such expectation values can be measured by repeatedly preparing \ensuremath∣\ensuremathψ⟩\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath∣\ensuremathψ⟩ and coupling the system to an apparatus. For this method, the precision of the measured value scales as 1N\frac{1}{\sqrt{N}}N​1​ for NNN repetitions of the experiment. For the problem of estimating the parameter \ensuremathϕ\ensuremath{\phi}\ensuremathϕ in an evolution e\ensuremath−i\ensuremathϕH{e}^{\ensuremath{-}i\ensuremath{\phi}H}e\ensuremath−i\ensuremathϕH, it is possible to achieve precision 1N\frac{1}{N}N1​ [the quantum metrology limit; see Giovannetti et al., Phys. Rev. Lett. 96, 010401 (2006)] provided that sufficient information about HHH and its spectrum is available. We consider the more general problem of estimating expectations of operators AAA with minimal prior knowledge of AAA. We give explicit algorithms that approach precision 1N\frac{1}{N}N1​ given a bound on the eigenvalues of AAA or on their tail distribution. These algorithms are particularly useful for simulating quantum systems on quantum computers because they enable efficient measurement of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of \ensuremath∣\ensuremathψ⟩\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath∣\ensuremathψ⟩ and U\ensuremath∣\ensuremathψ⟩U\ensuremath{\mid}\ensuremath{\psi}⟩U\ensuremath∣\ensuremathψ⟩, where UUU is an implementable unitary operator. We explicitly consider the issue of confidence levels in measuring observables and overlaps and show that, as expected, confidence levels can be improved exponentially with linear overhead. We further show that the algorithms given here can typically be parallelized with minimal increase in resource usage.
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.
It is suggested that computers exploiting quantum phenomena can solve tasks with an efficiency surpassing that achievable by classical computers. One example of this quantum advantage arises in the task of phase estimation. Here, an unknown eigenvalue of a unitary evolution is estimated, a critical subroutine for many quantum algorithms and quantum metrology protocols. For a fixed amount of resources, the utilisation of quantum coherence and entanglement can lead to a quadratic increase in precision of the final estimate compared to classical statistical limits. In systems affected by depolarising noise, including current quantum hardware, this improvement becomes a constant factor. While numerous phase estimation algorithms have been proposed claiming to achieve these theoretical limits, the lack of well-defined criteria for both precision and resource utilisation casts doubt on these results. Further issues arise due to a lack of point identification (the existence of multiple equally likely estimates) that some algorithms have. I aim to address these issues in this thesis. I employ statistical inference, the theoretical foundation for interpreting any quantum measurement, in order to find rigorous definitions for precision and resource use. By combining these definitions with previous work in the field of quantum metrology, I am able to derive theoretical precision limits that one can hope to achieve. I then outline several phase estimation algorithms that can achieve these limits in different contexts. These algorithms that I describe are split into three major groups. Algorithms in the first group focus on the process of point identification; one set of measurements provides a precise estimate of the unknown phase before an additional set of measurements removes any ambiguities present. This second set of measurements can be either decided before any measurement is taken or during measurements in an iterative fashion. I show that the balancing of resources used in each set of measurements can lead to optimally scaling precision both with and without depolarising noise. These algorithms are based upon two of my peer-reviewed articles, [Phys. Rev. A 106, 062615 (2022)] and [Phys. Rev. A 109, 042412 (2024)]. The second group of algorithms that I introduce use the principles of Bayesian experimental design. Here, previous measurement outcomes are used to select future measurements recursively. These algorithms require large classical computing times to select each measurement but boast high precision in certain metrics. The final group of algorithms exploit the oscillating nature of phases by using a discrete Fourier transform (DFT). The DFT can be implemented using either classical or quantum hardware, the latter of which leads to the famous quantum phase estimation algorithm (QPEA). In this thesis, I show that the QPEA performs worse than some classical precision limits. However, I also suggest some modifications to the QPEA circuit that can lead to optimal precision scaling. These results and modifications are described in one of my articles awaiting publication [arXiv:2505.00764].
We introduce an approximate description of an NNN-qubit state, which contains sufficient information to estimate the expectation value of any observable with precision independent of NNN. We show, in fact, that the error in the estimation of the observables' expectation values decreases as the inverse of the square root of the number of the system's identical preparations and increases, at most, linearly in a suitably defined, NNN-independent, seminorm of the observables. Building the approximate description of the NNN-qubit state only requires repetitions of single-qubit rotations followed by single-qubit measurements and can be considered for implementation on today's Noisy Intermediate-Scale Quantum (NISQ) computers. The access to the expectation values of all observables for a given state leads to an efficient variational method for the determination of the minimum eigenvalue of an observable. The method represents one example of the practical significance of the approximate description of the NNN-qubit state. We conclude by briefly discussing extensions to generative modelling and with fermionic operators.
In quantum mechanics, measuring the expectation value of a general observable has an inherent statistical uncertainty that is quantified by variance or mean squared error of measurement outcome. While the uncertainty can be reduced by averaging several samples, the number of samples should be minimized when each sample is very costly. This is especially the case for fault-tolerant quantum computing that involves measurement of multiple observables of nontrivial states in large quantum systems that exceed the capabilities of classical computers. In this work, we provide an adaptive quantum algorithm for estimating the expectation values of M general observables within root mean squared error ε simultaneously, using O ( ε − 1 M log  M ) queries to a state preparation oracle of a target state. This remarkably achieves the scaling of Heisenberg limit 1 / ε , a fundamental bound on the estimation precision in terms of mean squared error, together with the sublinear scaling of the number of observables M . The proposed method is an adaptive version of the quantum gradient-estimation algorithm and has a resource-efficient implementation due to its adaptiveness. Specifically, the space overhead in the proposed method is O ( M ) , which is independent from the estimation precision ε unlike noniterative algorithms. In addition, our method can avoid the numerical instability problem for constructing quantum circuits in a large-scale task (e.g., ε ≪ 1 in our case), which appears in the actual implementation of many algorithms relying on quantum signal processing techniques. Our method paves a new way to precisely understand and predict various physical properties in complicated quantum systems using quantum computers.
Classical simulability of quantum circuits plays a central role in characterizing and quantifying quantum computational advantage. Typically, the presence of non-Clifford gates introduces an exponential runtime overhead for classically estimating expectation values of observables. In this work, we introduce an efficient simulation framework that integrates the sum-over-Clifford method with quasi-probability-based stabilizer simulation to directly estimate expectation values. To optimize efficiency, we incorporate a circuit-adaptive reduction via variance estimation protocol, which terminates simulations once an empirical confidence bound is sufficient to guarantee the prescribed error tolerance. Our method circumvents overly conservative Hoeffding's inequality, thereby reducing the dependence on the gate-wise stabilizer extent from quadratic to linear in the low-variance regime, while maintaining the same space complexity as previous approaches. We explicitly prove the sample-complexity reduction for random quantum circuits and T-doped Clifford circuits and also empirically validate these advantages in the quantum approximate optimization algorithm and quantum kernel method.
Optimal and finite positive operator valued measurements on a finite number NNN of identically prepared systems have recently been presented. With physical realization in mind, we propose here optimal and minimal generalized quantum measurements for two-level systems. We explicitly construct them up to N=7N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}7N=7 and verify that they are minimal up to N=5N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5N=5.