%0 Journal Article %J ACM Transactions on Quantum Computing %D 2023 %T Quantum algorithm for estimating volumes of convex bodies %A Shouvanik Chakrabarti %A Andrew M. Childs %A Shih-Han Hung %A Tongyang Li %A Chunhao Wang %A Xiaodi Wu %X

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 O~(n3.5+n2.5/ε) queries to a membership oracle and O~(n5.5+n4.5/ε) additional arithmetic operations. For comparison, the best known classical algorithm uses O~(n4+n3/ε2) queries and O~(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.

%B ACM Transactions on Quantum Computing %V 4 %8 4/2023 %G eng %U https://arxiv.org/abs/1908.03903 %& 20