Topological Quantum Computation

This pedagogical introduction to topological quantum computation includes the following parts. First we provide an introduction to anyons and topological models. In particular we consider the properties of anyons and their relation to topological quantum computation. Then we present the quantum double models. These are stabiliser codes, that can be described very much like quantum error correcting codes. They include the toric code and various Abelian and non-Abelian extensions. Next the Jones polynomials are presented, which are topological invariants of links and knots that are related to anyons. Their evaluations by classical algorithms is computationally complex, but their approximation by quantum algorithms is efficient. Finally, we presenter an overview of the current state of topological quantum computation and present some open questions.

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