Topological Quantum Computation, Anyons and non-Abelian Gauge Potentials

This thesis deals with the study of topological quantum computation and the possible realization of non-Abelian anyons in cold atomic gases. Two main topics are investigated: the first subject is the quantum hashing technique to approximate unitary operators by braiding non-Abelian anyons, the second one is the analysis of systems of multicomponent ultracold atoms in the presence of an effective non-Abelian gauge potential giving rise to a quantum Hall regime. The common frame of these topics is the emergent study of topological phases of matters, driven by the necessity to overcome the Landau-Ginzburg paradigm to describe strongly correlated quantum systems such as the quantum Hall ones. To achieve this goal it is crucial to involve seemingly distant branches of knowledge such as conformal field theories, topological field theories, integrable models, knot theory, tensor category theory but also quantum information and computation, in order to deepen our understanding of the new and exciting experimental and numerical results given by the analysis of different systems sharing these topological properties.