Bosonsampling is far from uniform

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” from a uniform random outcome, at least “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” so strange 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 mistaken 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 directly 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 FermionSampling is solvable in classical polynomial time, and we reuse techniques from our BosonSampling analysis to characterize random FermionSampling distributions. 1 Background BosonSampling [1] can be defined as the following computational problem. We are given as input an m × n complex matrix A ∈ Cm×n (m ≥ n), whose n columns are orthonormal vectors in Cm. Let Φm,n be the set of all lists S = (s1, . . . , sm) of m nonnegative integers summing to n (we call these lists “experimental outcomes”); note that |Φm,n| = m+n−1 n ) . For each outcome S ∈ Φm,n, let AS ∈ Cn×n be the n× n matrix that consists of s1 copies of A’s first row, s2 copies of A’s second row, and so on. Then let DA be the following probability distribution over Φm,n: Pr DA [S] = |Per (AS)| s1! · · · sm! , (1) MIT. Email: [email protected]. This material is based upon work supported by the National Science Foundation under Grant No. 0844626, an Alan T. Waterman Award, a Sloan Fellowship, a TIBCO Chair, and an NSF STC on Science of Information. MIT. Email: [email protected]. Supported by an NSF Graduate Fellowship.