One of the crown jewels of complexity theory is Valiant’s 1979 theorem that computing the permanent of an n × n matrix is #P-hard. Here we show that, by using the model of linearoptical quantum computing—and in particular, a universality theorem due to Knill, Laflamme, and Milburn—one can give a different and arguably more intuitive proof of this theorem.

Let p > 2 be a prime, The group ring RG is calculated nearly up to Morita equivalence: The projections of RG into the simple components of KG are given explicitly and the endomorphism rings and homomorphism bimodules between the projective indecomposable RG-lattices are described.