Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants

Given a convex function fcolonℝd→ℝ, the problem of sampling from a distribution propto e-f(x) is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants intℝde-f(x)d x. First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number kappa and dimension d) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error ε. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity widetildeO(kappa1/2d) and widetildeO(kappa1/2d3/2/ε) for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in kappa,d,ε over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a 1/ε1-o(1) quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in ε.