Quantum Algorithms For Virtual Jones Polynomials Via Thistlethwaite Theorems

Recently a quantum algorithm for the Jones polynomial of virtual links was proposed by Kauffman and Dye via the implementation of the virtual braid group in anyonic topological quantum computation when the virtual crossings are considered as generalized swap gates. Also recently, a mathematical method for the computation of the Jones polynomial of a given virtual link in terms of the relative Tuttle polynomial of its face (Tait) graph with some suitable variable substitutions was proposed by Diao and Hetyei. The method of Diao and Hetyei is offered as an alternative to the ribbon graph approach according to which the Tutte polynomial of a given virtual link is computed in terms of the Bollobas-Riordan polynomial of the corresponding ribbon graph. The method of Diao and Hetyei can be considered as an extension of the celebrated Thistlethwaite theorem according to which invariant polynomials for knots and links are derived from invariant polynomials for graphs. Starting from these ideas we propose a quantum algorithm for the Jones polynomial of a given virtual link in terms of the generalized Tutte polynomials by exploiting the Thistlethwaite theorem and the Kauffman algorithm. Our method is claimed as the quantum version of the Diao-Hetyei method. Possible supersymmetric implementations of our algortihm are discussed jointly with its formulations using topological quantum lambda calculus.