A faster hafnian formula for complex matrices and its benchmarking on a supercomputer

Abstract

We introduce new and simple algorithms for the calculation of the number of perfect matchings of complex weighted, undirected graphs with and without loops. Our compact formulas for the hafnian and loop hafnian of $n \times n $ complex matrices run in $O(n^3 2^{n/2})$ time, are embarrassingly parallelizable and, to the best of our knowledge, are the fastest exact algorithms to compute these quantities. Despite our highly optimized algorithm, numerical benchmarks on the Titan supercomputer with matrices up to size $56 \times 56$ indicate that one would require the 288000 CPUs of this machine for about a month and a half to compute the hafnian of a $100 \times 100$ matrix.