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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 Dx 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 Ax 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 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.
1 Introduction
The Extended Church-Turing Thesis (ECT) says that all computational problems that are feasibly
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