Physics-inspired derivations of some algorithms for computing the permanent

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.