Learning quantum many-body systems from a few copies

Abstract

Estimating physical properties of quantum states from measurements is one of the most fundamental tasks in quantum science. In this work, we identify conditions on states under which it is possible to infer the expectation value of all quasi-local observables of a given locality up to a relative error from a number of samples that grows polylogarithmically with system size and polynomially on the locality of the target observables. This constitutes an exponential improvement over known tomography methods in some regimes. We achieve our results by combining one of the most well-established techniques to learn quantum states, the maximum entropy method, with techniques from the emerging fields of quantum optimal transport and classical shadows. We conjecture that our condition holds for all states exhibiting some form of decay of correlations and establish it for several subsets thereof. These include widely studied classes of states such as one-dimensional thermal and gapped ground states and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs. Moreover, we show improvements of the maximum entropy method beyond the sample complexity of independent interest. These include identifying regimes in which it is possible to perform the postprocessing efficiently and novel bounds on the condition number of covariance matrices of many-body states.Comment: 37 pages, 3 figure