The goal of this thesis is to examine some of the ways in which we might optimise the design of topological qubits. The topological operations which are imposed on qubits, in order to perform logic gates for topological quantum computations, are governed by the exchange group of the constituent particles. We examine representations of these exchange groups and investigate what restrictions their structure places on the effciency, reliability and universality of qubits (and multi-qubit systems) as a function of the number of particles composing them. Specific results are given for the limits placed on d-dimensional qudits where logic gates are imposed by braiding anyons in 2+1 dimensions. We also study qudits designed from ring-shaped, anyon-like excitations in 3+1 dimensions, where logic gates are implemented by elements of the loop braid group. We introduce the concept of local representations, where the generators of the loop braid group are defined to act non-trivially only on the local vector spaces associated with the rings which undergo the motion. We present a method for obtaining local representations of qudits and show how any such representation can be decomposed into representations which come from the quantum doubles of groups. Due to the dimension of the local representation being related to the number of generators, any non-Abelian properties of the representation are not compromised with an addition of extra operations, we conclude that universal representations may be easier to find than in previously discussed cases (though not for topological operations alone). We model a ring of Ising anyons in a fractional quantum Hall uid to study how interactions in a real environment may impact any qubits we have created. Fractional quantum Hall liquids are currently one of the most promising possibilities for the physical realisation of TQC and so present a natural choice of system in which to study these effects. We show how interactions between the anyons compromise the practicality of qubits defined by the fusion channels of anyon pairs and explore the use of the fermion number parity sectors as qubit states. Interactions between the anyon ring and the edge of the liquid are modelled to study the effect they will have on the state of the qubit. We perform numerical simulations, for a small system, to give some indication of how the edge interaction will in uence the reliability of the qubit.