Quantum algorithm for estimating volumes of convex bodies · arXivDesk
1908.03903Aug 11, 201961 pages, 8 figures. v2: Quantum query complexity improved to $\tilde{O}(n^{3}+n^{2.5}/ε)$ and number of additional arithmetic operations improved to $\tilde{O}(n^{5}+n^{4.5}/ε)$. v3: Improved Section 4.3.3 on nondestructive mean estimation and Section 6 on quantum lower bounds; various minor changes
Quantum algorithm for estimating volumes of convex bodies
Shouvanik Chakrabarti, Andrew M. Childs, Shih-Han Hung, Tongyang Li, Chunhao Wang, Xiaodi Wu
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+n2.5/ε)
Nearby in the stack
queries to a membership oracle and
O~(n5+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. To complement our quantum algorithms, we also prove that volume estimation requires
Ω(n+1/ε)
quantum membership queries, which rules out the possibility of exponential quantum speedup in