Topological Quantum Information Theory

This paper is an introduction to relationships between quantum topology and quantum computing. In this paper we discuss unitary solutions to the Yang-Baxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knottheoretic recoupling theory, its relationship with topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten-Reshetikhin-Turaev invariant of three manifolds.