Efficiently estimating quantum thermal properties from exponentially fewer samples

Abstract

We present an efficient protocol for estimating $M$ observables to precision $\epsilon$ from only $\mathcal{O}(\log (M)/{\epsilon }^2)$ copies of a Gibbs state $\rho \propto \exp (-\beta H)$, given access to its Hamiltonian $H$. The protocol uses single-copy, nonadaptive measurements and a total Hamiltonian simulation time of $\tilde{\mathcal{O}}(\beta M/{\epsilon }^2)$; we show that the sample complexity is optimal in a black-box setting. The key idea of our protocol is a new interpretation of quantum Gibbs samplers as detailed-balanced measurement channels: measurements that preserve the Gibbs state when outcomes are marginalized, and thus enable Gibbs samples to be reused for measuring subsequent observables. Consequently, shadow tomography of thermal states admits a general efficient algorithm when controlled Hamiltonian simulation is available, substantially boosting the data acquisition rate in quantum thermal simulation workflows.

arXiv comment24 pages, 2 figures, small corrections, clarifications, and shaving off a log factor in the gate complexity