The complexity of the fermionant, and immanants of constant width

@article{Mertens2013TheCO,
  title={The complexity of the fermionant, and immanants of constant width},
  author={Stephan Mertens and Cristopher Moore},
  journal={Theory of Computing},
  year={2013},
  volume={9},
  pages={273-282}
}
In the context of statistical physics, Chandrasekharan and Wiese recently introduced the fermionant Fermk, a determinant-like quantity 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 Turing reductions for any constant k > 2, and is ⊕P-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 Immλ A as a… CONTINUE READING