Efficiently estimating quantum thermal properties from exponentially fewer samples

We present an efficient protocol for estimating $M$ observables to precision $\varepsilon$ from only $\mathcal{O}(\log (M)/\varepsilon^2)$ copies of a Gibbs state $ρ\propto\exp(-βH)$, given access to its Hamiltonian $H$. The protocol uses single-copy, nonadaptive measurements and a total Hamiltonian