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.
