Learning quantum Hamiltonians from high-temperature Gibbs states and real-time evolutions

Abstract

The behaviour of a system is determined by its Hamiltonian. In many cases, the exact Hamiltonian is not known and has to be extracted by analysing the outcome of measurements. We study the problem of learning a local Hamiltonian H to a given precision, supposing either we are given copies of its Gibbs state ρ = e−βH/Tr(e−βH) at a known inverse temperature β or we have access to unitary real-time evolution e−itH for a known evolution time t. Improving on recent results, we show how to learn the coefficients of a local Hamiltonian H to error ε with S = O(logN/(βε)2) Gibbs states or with Q = O(logN/(tε)2) runs of the real-time evolution, where N is the number of qubits in the system and if β < βc and t < tc for some critical inverse temperature βc and critical evolution time tc. We design a classical post-processing algorithm with time complexity linear in the sample size in both cases, namely, O(NS) and O(NQ). In the Gibbs-state input case, we prove a matching lower bound, showing that our algorithm’s sample complexity is optimal, and hence, our time complexity is also optimal. Complexity of learning Hamiltonians from Gibbs states is an important issue for both many-body physics and machine learning. The optimal sample and time complexities of quantum Hamiltonian learning for high temperature has now been proven.