A faster calculation of Franck-Condon factors and Fock matrix elements of Gaussian unitaries using loop hafnians

We show that the Franck-Condon Factor (FCF) associated to a transition between initial and final vibrational states in two different potential energy surfaces, having $N$ and $M$ vibrational quanta, respectively, is equivalent to calculating the number of perfect matchings of a weighted graph with loops that has $P = N+M$ vertices. This last quantity is the loop hafnian of the (symmetric) adjacency matrix of the graph which can be calculated in $O(P^3 2^{P/2})$ steps. In the limit of small numbers of vibrational quanta per normal mode our loop hafnian formula significantly improves the speed at which FCFs can be calculated. Our results more generally apply to the calculation of the matrix elements of a bosonic Gaussian unitary between two multimode Fock states having $N$ and $M$ photons in total and provide a useful link between certain calculations of quantum chemistry, quantum optics and graph theory.