 
Summary: The Equivalence of Sampling and Searching
Scott Aaronson #
Abstract
In a sampling problem, we are given an input x # {0, 1} 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 # {0, 1} 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 RS such that, if C is any
``reasonable'' complexity class, then RS is in the search version of C if and only if S is in the
sampling version. What makes this nontrivial is that the same RS works for every C.
As an application, we prove the surprising result that SampP = SampBQP if and only
if FBPP = FBQP. In other words, classical computers can e#ciently sample the output
distribution of every quantum circuit, if and only if they can e#ciently solve every search
problem that quantum computers can solve.
1 Introduction
The Extended ChurchTuring Thesis (ECT) says that all computational problems that are feasibly
solvable in the physical world are feasibly solvable by a probabilistic Turing machine. By now,
there have been almost two decades of discussion about this thesis, and the challenge that quantum
computing poses to it. This paper is about a related question that has attracted surprisingly little
