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Matrix permanents arise naturally in the context of linear optical networksfed with nonclassical states of light. In this letter we tie the computationalcomplexity of a class of multi-dimensional integrals to the permanents of largematrices using a simple quantum optics argument. In this way we prove thatevaluating integrals in this class is \textbf{\#P}-hard. Our work provides anew approach for using methods from quantum physics to prove statements incomputer science.

A Quantum Optics Argument for the #P-hardness of a Class of Multidimensional Integrals