Efficient Shadow Tomography of Thermal States

Abstract

We present a general protocol for estimating $M$ observables from only $\mathcal{O}(\log (M)/{\epsilon }^2)$ copies of a Gibbs state whose Hamiltonian is accessible. The protocol uses single-copy, nonadaptive measurements and uses a total Hamiltonian simulation time of $\tilde{\mathcal{O}}({\hat{I}}^{2}M/{\epsilon }^2)$; 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.

20 pages, 1 figure