The Equivalence of Sampling and Searching

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 $\mathcal{D}_{x}$ over $\operatorname*{poly}\left( n\right)$-bit 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.