



@article {2448,
	title = {Quantum algorithm for estimating volumes of convex bodies},
	journal = {ACM Transactions on Quantum Computing},
	volume = {4},
	year = {2023},
	month = {4/2023},
	chapter = {20},
	abstract = {<p>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 \&quot;Chebyshev cooling\&quot;, 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.</p>},
	url = {https://arxiv.org/abs/1908.03903},
	author = {Shouvanik Chakrabarti and Andrew M. Childs and Shih-Han Hung and Tongyang Li and Chunhao Wang and Xiaodi Wu}
}
