BosonSampling is far from uniform

Abstract

BosonSampling, which we proposed three years ago, is a scheme for using linear-optical networks to solve sampling problems that appear to be intractable for a classical computer. \ In a recent manuscript, Gogolin et al.\ claimed that even an ideal BosonSampling device's output would be operationally indistinguishable\textquotedblright\ from a uniform random outcome, at least \textquotedblleft without detailed a priori knowledge; or at any rate, that telling the two apart might itself be a hard problem. We first answer these claims---explaining why the first is based on a definition of a priori knowledge such that, were it adopted, almost no quantum algorithm could be distinguished from a pure random-number source; while the second is neither new nor a practical obstacle to interesting BosonSampling experiments.However, we then go further, and address some interesting research questions inspired by Gogolin et al.'s arguments. We prove that, with high probability over a Haar-random matrix $A$, the BosonSampling distribution induced by $A$ is far from the uniform distribution in total variation distance. More surprisingly, and counter to Gogolin et al., we give an efficient algorithm that distinguishes these two distributions with constant bias. Finally, we offer three bonus results about BosonSampling. First, we report an observation of Fernando Brandao: that one can efficiently sample a distribution that has large entropy and that's indistinguishable from a BosonSampling distribution by any circuit of fixed polynomial size. Second, we show that BosonSampling distributions can be efficiently distinguished from uniform even with photon losses and for general initial states. Third, we offer the simplest known proof that Fermion Sampling is solvable in classical polynomial time, and we reuse techniques from our Boson Sampling analysis to characterize random FermionSampling distributions.