# Physics-inspired derivations of some algorithms for computing the permanent

## TL;DR

This paper introduces physics-inspired methods for computing the matrix permanent, using Grassmann integrals and fermion problems, leading to new derivations and estimators that connect to existing algorithms.

## Contribution

It formulates the permanent computation as a Grassmann integral, deriving known and new algorithms through physics-inspired techniques.

## Key findings

- Derivation of the Godsil-Gutman and Karmarkar estimators from Grassmann integrals.
- Connection of the permanent computation to interacting many-fermion problems.
- Introduction of new estimators and formulas based on gauge invariance.

## Abstract

We provide physics-inspired derivations of a number of algorithms for computing the permanent of a matrix. In particular we formulate the computation of the permanent as a Grassmann integral that may be viewed as an interacting many-fermion problem. Applying a discrete Hubbard-Stratonovich decoupling then gives approximation schemes that are equivalent to the familiar determinant Monte Carlo algorithm. This leads to elementary derivations of the well-known estimators of Godsil-Gutman and Karmarkar et al. Another straightfoward manipulation of the Grassmann integral, making use of gauge invariance, gives the efficient exact formula of Glynn. In addition to these known results we also give some additional estimators and formulas that are natural in our formulation.