Abstract:
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 \mathcal{D}_{x}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 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 \mathcal{C} is any “reasonable” complexity class, then R S is in the search version of \mathcal{C} if and only if S is in the sampling version. What makes this nontrivial is that the same R S works for every \mathcal{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.

Description:
6th International Computer Science Symposium in Russia, CSR 2011, St. Petersburg, Russia, June 14-18, 2011. Proceedings