Many quantum algorithms involve the evaluation of expectation values. Optimal strategies for estimating a single expectation value are known, requiring a number of state preparations that scales with the target error $ϵ$ as $\mathcal{O}(1/ϵ)$. In this Letter, we address the task of estimating the expectation values of $M$ different observables, each to within additive error $ϵ$, with the same $1/ϵ$ dependence. We describe an approach that leverages Gilyén et al.’s quantum gradient estimation algorithm to achieve $\mathcal{O}(\sqrt{M}/ϵ)$ scaling up to logarithmic factors, regardless of the commutation properties of the $M$ observables. We prove that this scaling is worst-case optimal in the high-precision regime if the state preparation is treated as a black box, even when the operators are mutually commuting. We highlight the flexibility of our approach by presenting several generalizations, including a strategy for accelerating the estimation of a collection of dynamic correlation functions.

• Received 19 January 2022
• Revised 30 August 2022
• Accepted 30 September 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.240501

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society