Gibbs Sampling of Periodic Potentials on a Quantum Computer

Motivated by applications in machine learning, we present a quantum algorithm for Gibbs sampling from continuous real-valued functions defined on high dimensional tori. We show that these families of functions satisfy a Poincar\'e inequality. We then use the techniques for solving linear systems and partial differential equations to design an algorithm that performs zeroeth order queries to a quantum oracle computing the energy function to return samples from its Gibbs distribution. 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. To this end, we develop provably efficient quantum algorithms for manipulating real analytic periodic functions.