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

## TL;DR

This paper introduces efficient algorithms for computing matrix permanents, especially for sparse and limited bandwidth matrices, enabling faster simulations of quantum systems like Boson sampling and exploring conjectures about random matrix permanents.

## Contribution

It presents new parallel and specialized algorithms for permanent computation, achieving exponential speed-up and linear time for certain matrices, and provides empirical analysis of permanent distributions in random matrices.

## Key findings

- Achieved a new world record for matrix size in permanent computation.
- Found good agreement with conjectures on permanent anti-concentration.
- Identified deviations from bounds in Boson sampling systems.

## 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 noticable deviations from a propose strengthening of a bound for the number of photons in a Boson sampling system.