Nearly Optimal Quantum Algorithm for Estimating Multiple Expectation Values

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 e as O(1=e thorn . In this Letter, we address the task of estimating the expectation values of M different observables, each to within additive error e, with the same 1=e dependence. We describe an approach that leverages Gilyen et al.'s quantum gradient estimation algorithm to achieve O( p =e thorn scaling up to ffiffiffiffiM 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.

Automated summary

This Letter addresses the task of estimating the expectation values of multiple observables in quantum algorithms. The authors propose an approach that utilizes a quantum gradient estimation algorithm to achieve efficient estimation. They demonstrate that this approach provides optimal scaling in the high-precision regime and showcases its flexibility.