We will consider the problem of learning the Hamiltonian of a quantum many-body system given samples from its Gibbs (thermal) state. The classical analog of this problem, known as learning graphical models or Boltzmann machines, is a well-studied question in machine learning and statistics. This talk will describe a sample-efficient algorithm for the quantum Hamiltonian learning problem at all constant temperatures. In particular, we prove that polynomially many samples in the number of particles (qudits) are necessary and sufficient for learning the parameters of a spatially local Hamiltonian in l_2-norm. Our main contribution is in establishing the strong convexity of the log-partition function of quantum many-body systems. In the process, we prove a lower bound on the variance of quasi-local operators with respect to the Gibbs state, which may be of independent interest.
