Non-abelian statistics of anyons in two dimensions have attracted considerable interest in the past 20 years, in part due to the potential for realising fault-tolerant quantum computation. In comparison, in three dimensions there exists a no-go theorem for point particles realising non-abelian statistics. However, three-dimensional condensed matter systems naturally support spatially extended excitations, such as loops, which can admit non-abelian statistics.
In this talk I will give a brief overview of topological quantum computing with anyons, utilising the connections between the mathematics of topological quantum field theories and Hamiltonian models of topological phases of matter. Building on this connection I will then discuss the ongoing work of describing non-abelian exchange statistics of loop excitations in three dimensions and the potential applications to quantum computation.