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
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 ϵ using Õ (n3+n2.5/ϵ) queries to a membership oracle and Õ (n5+n4.5/ϵ) additional arithmetic operations. For comparison, the best known classical algorithm uses Õ (n4+n3/ϵ2) queries and Õ (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 continuousspace quantum walks with rigorous bounds on discretization error.
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 Award ID(s):
 1816695
 NSFPAR ID:
 10172835
 Date Published:
 Journal Name:
 Annual Conference on Quantum Information Processing
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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n dimensional convex body within multiplicative error ε usingÕ(n^{3}+ n^{2.5}/ε ) queries to a membership oracle andÕ(n^{5}+n^{4.5}/ε) additional arithmetic operations. For comparison, the best known classical algorithm usesÕ(n^{3.5}+n^{3}/ε^{2}) queries andÕ(n^{5.5}+n^{5}/ε^{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 continuousspace quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requiresΩ (√ n+1/ε) quantum membership queries, which rules out the possibility of exponential quantum speedup inn and shows optimality of our algorithm in 1/ε up to polylogarithmic factors. 
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