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Summary: A Linear-Optical Proof that the Permanent is #P-Hard
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 #P-hard. 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 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 #P-hard, 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 exponentially-many nonnegative integers--and
#P-hard means "at least as hard as any #P problem."2,3
More concretely, Valiant gave a polynomial-time algorithm that takes as input an instance
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