In the previous chapters we introduced anyons and their properties, we presented how to perform topological quantum computation and studied several examples of topological models. There is a wide variety of research topics concerned with topological quantum computation. Among the many open questions, two have a singular importance. The first natural question is: which physical systems can support non-Abelian anyons? Realising non-Abelian anyons in the laboratory is of fundamental and practical interest. Such exotic statistical behaviour has not yet been encountered in nature. The physical realisation of non-Abelian anyons would be the first step towards the identification of a technological platform for the realisation of topological quantum computation. The second question concerns the efficiency of topological systems in combating errors. It has been proven that the effect of coherent environmental errors in the form of local Hamiltonian perturbations can be suppressed efficiently without degrading the topologically encoded information (Bravyi et al., 2010). Nevertheless, there is no mechanism that can protect topological order from incoherent probabilistic errors. Topological systems nevertheless constitute a rich and versatile medium that allows imaginative proposals to be developed (Chesi et al., 2010; Hamma et al., 2009).

Regarding the first question, we can identify two main categories of physical proposals for the realisation of two-dimensional topological systems: systems that are defined on the continuum and discrete systems defined on a lattice. It is natural to ask, which are the most promising architectures to realise in the laboratory?