The Equivalence of Sampling and Searching

TL;DR

This paper demonstrates the fundamental equivalence between sampling and search problems using Kolmogorov complexity, with implications for quantum computing and linear optics experiments.

Contribution

It establishes a formal equivalence between sampling and search problems within complexity theory, linking classical and quantum computational capabilities.

Findings

Sampling and search problems are equivalent under Kolmogorov complexity.

Classical sampling of quantum distributions is equivalent to classical solving of quantum search problems.

Linear-optics experiments can solve certain search problems that challenge classical computation.

Abstract

In a sampling problem, we are given an input x, and asked to sample approximately from a probability distribution D_x. In a search problem, we are given an input x, 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.