## The equivalence of sampling and searching

##### Author(s)

Aaronson, Scott
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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

##### Date issued

2011-06##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### 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

##### ISBN

978-3-642-20711-2

##### ISSN

0302-9743

1611-3349