Learning Quantum Hamiltonians at Any Temperature in Polynomial Time

Abstract

We study the problem of learning a local quantum Hamiltonian given copies of its Gibbs state = − /tr( − ) at a known inverse temperature > 0. Anshu, Arunachalam, Kuwahara, and Soleimanifar gave an algorithm to learn a Hamiltonian on qubits to precision with only polynomially many copies of the Gibbs state, but which takes exponential time. Obtaining a computationally e cient algorithm has been a major open problem, with prior work only resolving this in the limited cases of high temperature or commuting terms. We fully resolve this problem, giving a polynomial time algorithm for learning to precision from polynomially many copies of the Gibbs state at any constant > 0. Our main technical contribution is a new at polynomial approximation to the exponential function, and a translation between multi-variate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system. We then show that solving a low-degree sum-of-squares relaxation of this polynomial system su ces to accurately learn the Hamiltonian.