 
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 computingand in particular, a universality theorem due to Knill, Laflamme,
and Milburnone 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 problemsproblems that involve summing exponentiallymany nonnegative integersand
#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
