Quantum algorithm for estimating volumes of convex bodies

Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an n-dimensional convex body within multiplicative error ε using tildeO(n3+n2.5/ε) queries to a membership oracle and tildeO(n5+n4.5/ε) additional arithmetic operations. For comparison, the best known classical algorithm uses tildeO(n4+n3/ε2) queries and tildeO(n6+n5/ε2) additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of "Chebyshev cooling", where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requires Omega(√ n+1/ε) quantum membership queries, which rules out the possibility of exponential quantum speedup in n and shows optimality of our algorithm in 1/ε up to poly-logarithmic factors.