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16 April 2010Quantum 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 Bollobás-
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.
Mario Vélez andJuan Ospina
"Quantum algorithms for virtual Jones polynomials via Thistlethwaite theorems", Proc. SPIE 7702, Quantum Information and Computation VIII, 770206 (16 April 2010); https://doi.org/10.1117/12.849776
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Mario Vélez, Juan Ospina, "Quantum algorithms for virtual Jones polynomials via Thistlethwaite theorems," Proc. SPIE 7702, Quantum Information and Computation VIII, 770206 (16 April 2010); https://doi.org/10.1117/12.849776