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.
