The complexity of the fermionant, and immanants of constant width

In the context of statistical physics, Chandrasekharan and Wiese recently introduced the \emph{fermionant} ${\text{Ferm}}_k$, a determinant-like quantity where each permutation $\pi$ is weighted by $-k$ raised to the number of cycles in $\pi$. We show that computing ${\text{Ferm}}_k$ is #P-hard under Turing reductions for any constant $k>2$, and is $\text{oplusP}$-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 ${\text{Imm}}_{\lambda }A$ as a function of the Young diagram $\lambda$ in polynomial time, even if the width of $\lambda$ is restricted to be at most 2. In particular, if ${\text{Ferm}}_2$ is in P, or if ${\text{Imm}}_{\lambda }$ is in P for all $\lambda$ of width 2, then $\NP \subseteq \RP$ and there are randomized polynomial-time algorithms for NP-complete problems.