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We provide physics-inspired derivations of a number of algorithms forcomputing the permanent of a matrix. In particular we formulate the computationof the permanent as a Grassmann integral that may be viewed as an interactingmany-fermion problem. Applying a discrete Hubbard-Stratonovich decoupling thengives approximation schemes that are equivalent to the familiar determinantMonte Carlo algorithm. This leads to elementary derivations of the well-knownestimators of Godsil-Gutman and Karmarkar et al. Another straightfowardmanipulation of the Grassmann integral, making use of gauge invariance, givesthe efficient exact formula of Glynn. In addition to these known results wealso give some additional estimators and formulas that are natural in ourformulation.

Physics-inspired derivations of some algorithms for computing the permanent