  We describe the mathematical theory of topological quantum computing with
symmetry defects in the language of fusion categories and unitary
representations. Symmetry defects together with anyons are modeled by G-crossed
braided extensions of unitary modular tensor categories. The algebraic data of
these categories afford projective unitary representations of the braid group.
Elements in the image of such representations correspond to quantum gates
arising from exchanging anyons and symmetry defects in topological phases of
matter with symmetry. We provide some small examples that highlight features of
practical interest for quantum computing. In particular, symmetry defects show
the potential to generate non-abelian statistics from abelian topological
phases and to be used as as a tool to enlarge the set of quantum gates
accessible to an anyonic device.
