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 $D_{x}$ over poly(n)-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 and algorithmic information theory 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 C is any "reasonable" complexity class, then $R_{S}$ is in the search version of C if and only if S is in the sampling version.

As one application, we show 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. A second application is that, assuming a plausible conjecture, there exists a search problem R that can be solved using a simple linear-optics experiment, but that cannot be solved efficiently by a classical computer unless the polynomial hierarchy collapses. That application will be described in a forthcoming paper with Alex Arkhipov on the computational complexity of linear optics.

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 $D_{x}$ over poly(n)-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 and algorithmic

information theory 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 C is any "reasonable" complexity class, then $R_{S}$ is in the search

version of C if and only if S is in the sampling version.

As one application, we show 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. A second application is that, assuming some plausible

conjectures, there exists a search problem R that can be solved using a

simple linear-optics experiment, but that cannot be solved efficiently by a

classical computer unless the polynomial hierarchy collapses. That

application will be described in a forthcoming paper with Alex Arkhipov on the

computational complexity of linear optics.