A quantum computer must be capable of manipulating quantum information while at the same time protecting it from error and loss of quantum coherence due to interactions with the environment. Topological quantum computation (TQC) offers a particularly elegant way to achieve this. In TQC, quantum information is stored in exotic states of matter which are intrinsically protected from decoherence, and quantum computation is carried out by dragging particle-like excitations (quasiparticles) around one another in two space dimensions. The resulting quasiparticle trajectories define world-lines in three-dimensional space-time, and the corresponding computation depends only on the topology of the braids formed by the world-lines.

Quasiparticles that can be used for TQC are expected to exist in a variety of fractional quantum Hall states, among them the so-called Fibonacci anyons. These quasiparticles are conjectured to exist in the ν = 12/5 fractional quantum Hall state which has been observed in experiments. It has been shown that qubits can be encoded using three or four Fibonacci anyons and single-qubit gates can be carried out by braiding quasiparticles within each qubit. Braids that approximate single-qubit gates can be found through brute force searching and the result can be systematically improved, to any desired accuracy, by applying the Solovay-Kitaev algorithm in SU(2).

Two-qubit gates are significantly harder to implement, mostly due to the following two reasons. First, the Hilbert space of the quasiparticles forming two qubits is considerably larger than the Hilbert space of the quasiparticles of a single qubit. Therefore, performing a brute force search to find braids that approximate two-qubit gates, as well as the implementation of the Solovay-Kitaev algorithm for subsequent improvements are prohibitively more difficult. Second, to construct any entangling two-qubit gate, one needs to braid some of the quasiparticles from one qubit around quasiparticles of the other qubit. This process will inevitably lead to leakage errors, i.e. transitions from the qubit space to other available states in the Hilbert space.

In this thesis, I will present several efficient methods to construct two-qubit gates using a specific class of quasiparticles. In particular, I show that the problem of finding braids that correspond to two-qubit gates can be reduced to a series of smaller problems which involve braiding only three objects at a time. The required computational power for finding these braids is equivalent to that needed to find single-qubit gates, therefore, these braids can be found with the same high degree of accuracy and efficiency. The end result of this work is an efficient procedure for translating (or “compiling”) arbitrary quantum algorithms into specific braiding patterns for Fibonacci anyons, as well as quasiparticles of certain other fractional quantum Hall states that can be used for TQC.