  In contrast to the determinant, no algorithm is known for the exact
determination of the permanent of a square matrix that runs in time polynomial
in its dimension. Consequently, non interacting fermions are classically
efficiently simulatable while non-interacting bosons are not, underpinning
quantum supremacy arguments for sampling the output distribution of photon
interferometer arrays. This work introduces a graph-theoretic framework that
bridges both the determinant and permanent. The only non-zero eigenvalues of a
sparse non-Hermitian operator $\breve{M}$ for $n$ spin-$1/2$ particles are the
$n$th roots of the permanent or determinant of an $n\times n$ matrix $M$,
interpreting basis states as bosonic or fermionic occupation states,
respectively. This operator can be used to design a simple and straightforward
method for the classical determination of the permanent that matches the
efficiency of the best-known algorithm. Gauss-Jordan elimination for the
determinant of $M$ is then equivalent to the successive removal of the
generalized zero eigenspace of the fermionic $\breve{M}$, equivalent to the
deletion of some nodes and reweighting of the remaining edges in the graph such
that only $n$ nodes survive after the last step. In the bosonic case, the
successive removal of generalized zero eigenspaces for $\breve{M}$ is also
equivalent to node deletion, but new edges are added during this process, which
gives rise to the higher complexity of computing the permanent. Our analysis
may point the way to new strategies for classical and quantum evaluation of the
permanent.
