Provable Quantum--Classical Separation for Continuous Gibbs Sampling

Abstract

We prove the first quantum--classical separation for a sampling problem over a continuous domain. For a class of Gibbs states $p\propto e^{-\beta E}$ on the torus $\mathbb{T}^d$ with smooth ($s$-Gevrey) potential and barrier amplitude $\alpha=e^{\beta\Delta}$, where $\Delta = \max E-\min E$, every classical algorithm---querying the value, gradient, or any higher-order derivatives of the log-density---requires $\Omega(\alpha)$ queries to sample at constant accuracy in total variation distance, while a quantum algorithm based on quantum singular value thresholding and temperature annealing samples with $\tilde{O}\left(\sqrt{\alpha}\right)$ queries to an oracle for the gradient. The advantage is quadratic in the barrier amplitude, which becomes exponential in the dimension, $e^{\Omega(d)}$, at low temperature. The classical bound is information-theoretic, holding for every classical algorithm with query access to the Gibbs potential and its derivatives at any order.