Quantum Algorithms for the Jones Polynomial

TL;DR

This paper extends quantum algorithms for computing the Jones polynomial to continuous parameter ranges, generalizing the AJL algorithm and employing diagrammatic techniques for proof.

Contribution

It introduces a generalized quantum algorithm for the Jones polynomial applicable over continuous parameter ranges, expanding prior discrete methods.

Findings

Generalization of the AJL algorithm to continuous values.

Proofs using diagrammatic techniques.

Application potential in NMR quantum computation.

Abstract

This paper gives a generalization of the AJL algorithm and unitary braid group representation for quantum computation of the Jones polynomial to continuous ranges of values on the unit circle of the Jones parameter. We show that our 3-strand algorithm for the Jones polynomial is a special case of this generalization of the AJL algorithm. The present paper uses diagrammatic techniques to prove these results. The techniques of this paper have been used and will be used in the future in work with R. Marx, A. Fahmy, L. H. Kauffman, S. J. Lomonaco Jr.,A. Sporl, N. Pomplun, T. Schulte Herbruggen, J. M. Meyers, and S. J. Glaser on NMR quantum computation of the Jones polynomial.