A permanent formula for the Jones polynomial

Details A permanent formula for the Jones polynomial A permanent formula for the Jones polynomial

2007-05-31

arxiv.org/abs/0705.4548v3

Adv. in Appl. Math., 47 (2011) 659-667

Mathematics

Quantum Algebra

To appear in Advances in Applied Mathematics

Scientific paper

10.1016/j.aam.2011.03.003

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.

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