A permanent formula for the Jones polynomial

The permanent of a square matrix is defined in a way similar to the determinant, but without using signs. The exact computation of the permanent is hard, but there are Monte-Carlo algorithms that can estimate general permanents. Given a planar diagram of a link L with $n$ crossings, we define a 7n by 7n matrix whose permanent equals to the Jones polynomial of L. This result accompanied with recent work of Freedman, Kitaev, Larson and Wang provides a Monte-Carlo algorithm to any decision problem belonging to the class BQP, i.e. such that it can be computed with bounded error in polynomial time using quantum resources.Comment: To appear in Advances in Applied Mathematic

Identifying lens spaces in polynomial time

We show that if a closed, oriented 3-manifold M is promised to be homeomorphic to a lens space L(n,k) with n and k unknown, then we can compute both n and k in polynomial time in the size of the triangulation of M. The tricky part is the parameter k. The idea of the algorithm is to calculate Reidemeister torsion using numerical analysis over the complex numbers, rather than working directly in a cyclotomic field.Comment: 5 pages. A major revision with a new title, and with a classical algorithm rather than a quantum algorith

A Linear-Optical Proof that the Permanent is #P-Hard

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.Comment: 12 pages, 2 figures, to appear in Proceedings of the Royal Society A. doi: 10.1098/rspa.2011.023

Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy

We consider quantum computations comprising only commuting gates, known as IQP computations, and provide compelling evidence that the task of sampling their output probability distributions is unlikely to be achievable by any efficient classical means. More specifically we introduce the class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection, and prove first that post-IQP equals the classical class PP. Using this result we show that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, even up to 41% multiplicative error in the probabilities, then the infinite tower of classical complexity classes known as the polynomial hierarchy, would collapse to its third level. We mention some further results on the classical simulation properties of IQP circuit families, in particular showing that if the output distribution results from measurements on only O(log n) lines then it may in fact be classically efficiently sampled.Comment: 13 page