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A general quantum algorithm for knot and link polynomials

Jon Yard, Los Alamos National Laboratory

(Session 6 : Saturday from 09:15-9:45)

Abstract. In this talk, I will present a quantum algorithm for approximating topological invariants of knots and links coming from Markov traces on centralizer algebras of quantum groups. The method is based on a general formalism for efficiently implementing, on a quantum computer, representations of braid groups associated with path algebras. The general framework presented accommodates known quantum algorithms for approximately evaluating the Jones and HOMFLYPT polynomials - which arise from Markov traces on Temperley-Lieb and Hecke algebras associated to deformations of unitary groups. The framework also allows one to approximately evaluate the Kauffman polynomial invariants which arise from Markov traces on Birman-Wenzl-Murakami algebras associated to deformations of the orthogonal and symplectic groups. Time permitting, I will also comment on the cases in which approximating the Kauffman polynomial is a universal quantum algorithm which solves a Promise-BQP-complete problem.

Acknowledgements: This is joint work with Cris Moore (University of New Mexico, Santa Fe Institute).