Gibbs Sampling of Periodic Potentials on a Quantum Computer

Motivated by applications in machine learning, we present a quantum algorithm for Gibbs sampling from a continuous real-valued function defined on a high dimensional torus. Our algorithm relies on techniques for solving linear systems and partial differential equations and performs zeroeth order queries to a quantum oracle computing the energy function. We then analyze the query and gate complexity of our algorithm and prove that the algorithm has a polylogarithmic dependence on approximation error (in total variation distance) and a polynomial dependence on the number of variables, although it suffers from an exponentially poor dependence on temperature.