The complexity of the fermionant, and immanants of constant width

In the context of statistical physics, Chandrasekharan and Wiese recently introduced the fermionant Fermk, a determinant-like function of a matrix where each permutation π is weighted by −k raised to the number of cycles in π . We show that computing Fermk is #P-hard under polynomial-time Turing reductions for any constant k > 2, and is ⊕P-hard for k = 2, where both results hold even for the adjacency matrices of planar graphs. As a consequence, unless the polynomial-time hierarchy collapses, it is impossible to compute the immanant Immλ 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, unless NP⊆ RP, Ferm2 is not in P, and there are Young diagrams λ of width 2 such that Immλ is not in P. ACM Classification: F.2.1 AMS Classification: 68Q17