Quantum Computation of Jones’ Polynomials

It is one of the challenging problems to construct an efficient quantum algorithm which can compute the Jones’ polynomial for any knot or link obtained from closure or capping of a n-strand braid. We recapitulate the construction of braid-group Bn representations from vertex models. We perform orthogonal transformation involving quantum Clebsch-Gordan coefficient matrix on the qubit basis to obtain eigen basis for the braiding generators. Using the transformed bases, we propose a quantum algorithm to determine the Jones’ Polynomial for any knot or link.