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 ndimensional convex body within multiplicative error epsilon using O-similar to (n(3) + n(2.5)/is an element of) queries to a membership oracle and O-similar to (n(5) + n(4.5)/is an element of) additional arithmetic operations. For comparison, the best known classical algorithm uses O-similar to (n(3.5) + n(3)/is an element of(2)) queries and O-similar to (n(5.5) + n(5)/(2)) 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 techniqueswhen 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( root n + 1/is an element of) quantum membership queries, which rules out the possibility of exponential quantum speedup in n and shows optimality of our algorithm in 1/is an element of up to poly-logarithmic factors.