A LinearOptical Proof that the Permanent is #PHard Scott Aaronson # Summary: A Linear­Optical Proof that the Permanent is #P­Hard Scott Aaronson # For Les Valiant, on the occasion of his Turing Award Abstract One of the crown jewels of complexity theory is Valiant's 1979 theorem that computing the permanent of an n × n matrix is #P­hard. Here we show that, by using the model of linear­ optical quantum computing---and in particular, a universality theorem due to Knill, Laflamme, and Milburn---one can give a di#erent and arguably more intuitive proof of this theorem. 1 Introduction Given an n × n matrix A = (a i,j ), the permanent of A is defined as Per (A) = # ##Sn n # i=1 a i,#(i) . A seminal result of Valiant [15] says that computing Per (A) is #P­hard, if A is a matrix over (say) the integers, the nonnegative integers, or the set {0, 1}. 1 Here #P means (informally) the class of counting problems---problems that involve summing exponentially­many nonnegative integers---and #P­hard means ``at least as hard as any #P problem.'' 2,3 More concretely, Valiant gave a polynomial­time algorithm that takes as input an instance