The equivalence of sampling and searching
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Aaronson_The equivalence.pdf
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195.13 KB
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Author(s)
Aaronson, Scott
Date Issued
June 2011
Journal
Computer Science – Theory and Applications
Publisher
Springer Berlin/Heidelberg
Citation
Aaronson, Scott. “The Equivalence of Sampling and Searching.” Computer Science – Theory and Applications. Ed. Alexander Kulikov & Nikolay Vereshchagin. Vol. 6651. Lecture Notes in Computer Science: Springer Berlin Heidelberg, 2011. 1-14. Web. 9 Aug. 2012.
Version
Author's final manuscript
Abstract
In a sampling problem, we are given an input x\in\left\{ 0,1\right\} ^{n} , and asked to sample approximately from a probability distribution \mathcal{D}_{x}strings. In a search problem, we are given an input x\in\left\{ 0,1\right\} ^{n} , and asked to find a member of a nonempty set A x with high probability. (An example is finding a Nash equilibrium.) In this paper, we use tools from Kolmogorov complexity to show that sampling and search problems are “essentially equivalent.” More precisely, for any sampling problem S, there exists a search problem R S such that, if \mathcal{C} is any “reasonable” complexity class, then R S is in the search version of \mathcal{C} if and only if S is in the sampling version. What makes this nontrivial is that the same R S works for every \mathcal{C}.
As an application, we prove the surprising result that SampP = SampBQP if and only if FBPP = FBQP. In other words, classical computers can efficiently sample the output distribution of every quantum circuit, if and only if they can efficiently solve every search problem that quantum computers can solve.
Description
6th International Computer Science Symposium in Russia, CSR 2011, St. Petersburg, Russia, June 14-18, 2011. Proceedings
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/978-3-642-20712-9_1
