Photon-number moments and cumulants of Gaussian states

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We develop closed-form expressions for the moments and cumulants of Gaussian states when measured in the photon-number basis.

• We express the photon-number moments of a Gaussian state in terms of the loop Hafnian, a function that when applied to a $(0,1)$-matrix representing the adjacency of a graph, counts the number of its perfect matchings.
• Similarly, we express the photon-number cumulants in terms of the Montrealer, a newly introduced matrix function that when applied to a $(0,1)$-matrix counts the number of Hamiltonian cycles of that graph.
• Based on these graph-theoretic connections, we show that the calculation of photon-number moments and cumulants are #P-hard.