Efficient Shadow Tomography of Thermal States

TL;DR

This paper introduces an efficient protocol for estimating multiple observables from thermal states with minimal copies, leveraging a novel interpretation of Gibbs samplers to significantly reduce measurement costs.

Contribution

The authors develop a new shadow tomography protocol for thermal states that is optimal in sample complexity and uses a novel interpretation of Gibbs samplers as detailed-balance measurement channels.

Findings

Sample complexity is logarithmic in the number of observables.

The protocol uses nonadaptive, single-copy measurements.

It achieves optimality in a black-box setting.

Abstract

We present a general protocol for estimating $M$ observables from only $\mathcal{O}(\log (M)/\varepsilon^2)$ copies of a Gibbs state whose Hamiltonian is accessible. The protocol uses single-copy, nonadaptive measurements and uses a total Hamiltonian simulation time of $\widetilde{\mathcal{O}}(\beta M/\varepsilon^2)$; we show that the sample complexity is optimal in a black-box setting where exponential time Hamiltonian simulation is prohibited. The key idea is a new interpretation of quantum Gibbs samplers as \textit{detailed-balance measurement channels}: measurements that preserve the Gibbs state when outcomes are marginalized. Consequently, shadow tomography of thermal states admits a general efficient algorithm when the Hamiltonian is known, substantially lowering the readout cost in quantum thermal simulation.