  Matrix permanents arise naturally in the context of linear optical networks
fed with nonclassical states of light. In this letter we tie the computational
complexity of a class of multi-dimensional integrals to the permanents of large
matrices using a simple quantum optics argument. In this way we prove that
evaluating integrals in this class is \textbf{\#P}-hard. Our work provides a
new approach for using methods from quantum physics to prove statements in
computer science.
