Topological Quantum Information, Khovanov Homology and the Jones Polynomial

Details Topological Quantum Information, Khovanov Homology and the Jones Polynomial Topological Quantum Information, Khovanov Homology and the Jones Polynomial

2010-01-04

arxiv.org/abs/1001.0354v3

Mathematics

Geometric Topology

21 pages, 5 figures, LaTeX document

Scientific paper

In this paper we give a quantum statistical interpretation for the bracket polynomial state sum and for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to all values of the polynomial variable that lie on the unit circle in the complex plane. Letting C(K) denote the Hilbert space for this model, there is a natural unitary transformation U from C(K) to itself such that = where |F> is a sum over basis states for C(K). The quantum algorithm arises directly from this formula via the Hadamard Test. We then show that the framework for our quantum model for the bracket polynomial is a natural setting for Khovanov homology. The Hilbert space C(K) of our model has basis in one-to-one correspondence with the enhanced states of the bracket state summmation and is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. We show that for the Khovanov boundary operator d defined on C(K) we have the relationship dU + Ud = 0. Consequently, the unitary operator U acts on the Khovanov homology, and we therefore obtain a direct relationship between Khovanov homology and this quantum algorithm for the Jones polynomial. The formula for the Jones polynomial as a graded Euler characteristic is now expressed in terms of the eigenvalues of U and the Euler characteristics of the eigenspaces of U in the homology. The quantum algorithm given here is inefficient, and so it remains an open problem to determine better quantum algorithms that involve both the Jones polynomial and the Khovanov homology.

Affiliated with

Also associated with

No associations

Rating

Topological Quantum Information, Khovanov Homology and the Jones Polynomial does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Topological Quantum Information, Khovanov Homology and the Jones Polynomial, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Topological Quantum Information, Khovanov Homology and the Jones Polynomial will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-6424