A Bayesian analysis of classical shadows

TL;DR

This paper explores the relationship between classical shadows and Bayesian mean estimation in quantum state estimation, showing that Bayesian methods can improve accuracy and provide a formal statistical framework.

Contribution

It introduces a Bayesian perspective on classical shadows, including a pseudo-likelihood approach that maintains physical states and improves estimation accuracy.

Findings

Bayesian mean estimation often yields lower average error.

Classical shadows excel in high-fidelity state scenarios.

The pseudo-likelihood emulates classical shadows' advantages within a Bayesian framework.

Abstract

The method of classical shadows heralds unprecedented opportunities for quantum estimation with limited measurements [H.-Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. 16, 1050 (2020)]. Yet its relationship to established quantum tomographic approaches, particularly those based on likelihood models, remains unclear. In this article, we investigate classical shadows through the lens of Bayesian mean estimation (BME). In direct tests on numerical data, BME is found to attain significantly lower error on average, but classical shadows prove remarkably more accurate in specific situations -- such as high-fidelity ground truth states -- which are improbable in a fully uniform Hilbert space. We then introduce an observable-oriented pseudo-likelihood that successfully emulates the dimension-independence and state-specific optimality of classical shadows, but within a Bayesian framework that ensures only physical states. Our research reveals how classical shadows effect important departures from conventional thinking in quantum state estimation, as well as the utility of Bayesian methods for uncovering and formalizing statistical assumptions.