In some recent papers, e.g. Phys. Rev. Lett. 119, 170501, the problem of gaussian boson sampling is shown to be related to the hafnian of a certain matrix. When applied to the adjacency matrix of a graph, the hafnian gives the number of perfect mathings in that graph.

On the other hand, for some oriented graphs the number of perfect matchings is given by the pfaffian of the antisymmetric adjacency matrix.

Question: Is the pfaffian related to "fermion sampling"?

  • $\begingroup$ Yes. - - - - - - - $\endgroup$ Jun 9, 2021 at 0:06
  • 2
    $\begingroup$ @NorbertSchuch, perhaps a reference? $\endgroup$
    – thedude
    Jun 9, 2021 at 11:53
  • $\begingroup$ Computing multi-point correlators of fermions in Gaussian states uses Wick's theorem, which says that those correlators are given as the Pfaffian of the two-point correlation matrix. $\endgroup$ Jun 9, 2021 at 15:35
  • $\begingroup$ @NorbertSchuch Could you point us to a reference where this is discussed in more detail? $\endgroup$
    – Shasa
    Dec 14, 2022 at 13:06
  • 1
    $\begingroup$ @Shasa Anything on Slater determinants will do (plus a bit of own work -- but I'm sure someone also wrote something on fermion sampling). For instance, if you do Monte Carlo with Gaussian fermionic wavefunction (typically with some Gutzwiller projection on top), the way the Monte Carlo sampling works is that it needs ratios of probabilities, which are obtained as ratios of Pfaffians (which are, in fact, even much easier to sample than the Pfaffian itself, because the matrices differ only in a few entries). $\endgroup$ Dec 14, 2022 at 22:33


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