Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Parallel repetition in projection games and a concentration bound
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The complexity of computing a Nash equilibrium
Communications of the ACM - Inspiring Women in Computing
An Introduction to Kolmogorov Complexity and Its Applications
An Introduction to Kolmogorov Complexity and Its Applications
The computational complexity of linear optics
Proceedings of the forty-third annual ACM symposium on Theory of computing
Theoretical Computer Science
The computational complexity of linear optics
Proceedings of the forty-third annual ACM symposium on Theory of computing
Time hierarchies for sampling distributions
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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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.