Estimating Quantum Hamiltonians via Joint Measurements of Noisy Non-Commuting Observables

TL;DR

This paper introduces a joint measurement method for estimating expectation values of incompatible quantum observables, achieving comparable efficiency to classical shadows and improving performance in noisy measurement scenarios.

Contribution

The work presents a novel joint measurement approach for quantum expectation estimation, linking it to classical shadows and optimizing for noisy measurement conditions.

Findings

Method achieves estimation precision with fewer repetitions.

Performance matches classical shadow protocols in noiseless settings.

Adapts to noisy measurements, reducing sample complexity.

Abstract

Estimation of expectation values of incompatible observables is an essential practical task in quantum computing, especially for approximating energies of chemical and other many-body quantum systems. In this work we introduce a method for this purpose based on performing a single joint measurement that can be implemented locally and whose marginals yield noisy (unsharp) versions of the target set of non-commuting Pauli observables. We derive bounds on the number of experimental repetitions required to estimate energies up to a certain precision. We compare this strategy to the classical shadow formalism and show that our method yields the same performance as the locally biased classical shadow protocol. We also highlight some general connections between the two approaches by showing that classical shadows can be used to construct joint measurements and vice versa. Finally, we adapt the joint measurement strategy to minimise the sample complexity when the implementation of measurements is assumed noisy. This can provide significant efficiency improvements compared to known generalisations of classical shadows to noisy scenarios.