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 $epsilon$ using $tilde{O}(n^{3}+n^{2.5}/epsilon)$ queries to a membership oracle and $tilde{O}(n^{5}+n^{4.5}/epsilon)$ additional arithmetic operations. For comparison, the best known classical algorithm uses $tilde{O}(n^{4}+n^{3}/epsilon^{2})$ queries and $tilde{O}(n^{6}+n^{5}/epsilon^{2})$ 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 techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error.
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