Efficient computation of permanents, with applications to Boson sampling and random matrices

Abstract

In order to find the outcome probabilities of quantum mechanical systems like the optical networks underlying Boson sampling, it is necessary to be able to compute the permanents of unitary matrices, a computationally hard task. Here we first discuss how to compute the permanent efficiently on a parallel computer, followed by algorithms which provide an exponential speed-up for sparse matrices and linear run times for matrices of limited bandwidth. The parallel algorithm has been implemented in a freely available software package, also available in an efficient serial version. As part of the timing runs for this package we set a new world record for the matrix order on which a permanent has been computed.Next we perform a simulation study of several conjectures regarding the distribution of the permanent for random matrices. Here we focus on permanent anti-concentration conjecture, which has been used to find the classical computational complexity of Boson sampling. We find a good agreement with the basic versions of these conjectures, and based on our data we propose refined versions of some of them. For small systems we also find noticeable deviations from a proposed strengthening of a bound for the number of photons in a Boson sampling system.