 
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 different and arguably more intuitive proof of this theorem.
1 Introduction
Given an n × n matrix A = (ai,j), the permanent of A is defined as
Per (A) =
Sn
n
i=1
ai,(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
