The Hafnian Master Theorem

We present the Hafnian Master Theorem which is a hafnian's generalization of the classical Permanent Master Theorem of MacMahon. It discloses an easy-to-compute generating function for the hafnians related to a remarkable class of multivariate random variables whose sampling and probability density function are ♯P-hard for computing. The theorem provides a pathway to a unified analysis of the nature's ♯P-hard problems, including problems in quantum computing, computer science, theory of computational complexity, combinatorics, statistics, cryptography, theory of graphs, number theory, physics of many-body systems, critical phenomena, quantum field theory, etc.