The complexity of the fermionant, and immanants of constant width

Abstract

In the context of statistical physics, Chandrasekharan and Wiese recently introduced the \emph{fermionant} $\Ferm_k$, a determinant-like quantity where each permutation $π$ is weighted by $-k$ raised to the number of cycles in $π$. We show that computing $\Ferm_k$ is #P-hard under Turing reductions for any constant $k > 2$, and is $\oplusP$-hard for $k=2$, even for the adjacency matrices of planar graphs. As a consequence, unless the polynomial hierarchy collapses, it is impossible to compute the immanant $\Imm_λ\,A$ as a function of the Young diagram $λ$ in polynomial time, even if the width of $λ$ is restricted to be at most 2. In particular, if $\Ferm_2$ is in P, or if $\Imm_λ$ is in P for all $λ$ of width 2, then $\NP \subseteq \RP$ and there are randomized polynomial-time algorithms for NP-complete problems.