In topological quantum computation, quantum information is stored in states which are intrinsically protected from decoherence, and quantum gates are carried out by dragging particlelike excitations (quasiparticles) around one another in two space dimensions. The resulting quasiparticle trajectories define world lines in three-dimensional space-time, and the corresponding quantum gates depend only on the topology of the braids formed by these world lines. We show how to find braids that yield a universal set of quantum gates for qubits encoded using a specific kind of quasiparticle which is particularly promising for experimental realization.

A method for compiling quantum algorithms into specific braiding patterns for nonabelian quasiparticles described by the so-called Fibonacci anyon model is developed. The method is based on the observation that a universal set of quantum gates acting on qubits encoded using triplets of these quasiparticles can be built entirely out of three-stranded braids (three-braids). These three-braids can then be efficiently compiled and improved to any required accuracy using the Solovay-Kitaev algorithm.

We demonstrate numerically that non-Abelian quasihole (qh) excitations of the ν = 5/2 fractional quantum Hall state have some of the key properties necessary to support quantum computation. We find that as the qh spacing is increased, the unitary transformation which describes winding two qh's around each other converges exponentially to its asymptotic limit and that the two orthogonal wavefunctions describing a system with four qh's become exponentially degenerate. We calculate the length scales for these two decays to be ξU ≈ 2.7 0 and ξE ≈ 2.3 0 respectively. Additionally we determine which fusion channel is lower in energy when two qh's are brought close together.The proposal to use quantum Hall states as a platform for quantum computation has spurred a great deal of interest 1,2,3 . These quantum Hall systems are believed to have natural "topological" immunity to decoherence and therefore hold particular promise for quantum computation. In so-called non-Abelian quantum Hall systems, the ground state is highly degenerate in the presence of quasiparticles (qp's), and this degenerate space can be used to store quantum information. Operations on this space are then performed by adiabatically dragging qp's around each other, thus "braiding" their world-lines in 2+1 dimensions.Although there is currently no definitive experimental evidence that non-Abelian quantum Hall states even exist, the community now strongly suspects 1 that the quantum Hall plateau observed at Landau level (LL) filling fraction ν = 5/2 is the non-Abelian Moore-Read (MR) phase 4 (or its closely related particle-hole conjugate 5 ). While the MR phase is, strictly speaking, not capable of universal topological quantum computation (computation by braiding qp's around each other at large distances), a scheme has been devised 6 that in principle allows error free quantum computation by supplementing these topological processes with nontopological processes where qp's are moved together and allowed to interact. Furthermore, the MR phase is frequently viewed as the simplest paradigm of a non-Abelian state of matter, and is therefore a logical starting point for detailed analysis 1 . In order for topological (or partially topological) schemes for quantum computation to be scalable (i.e., to allow large scale quantum computation), a number of crucial conditions must hold 1 . Condition (1) As all of the qp's are moved apart from one another, the splitting of the energy levels of the putatively degenerate ground state space must converge to zero at least as fast as e −R/ξ E where R is the minimum distance between qp's. In the literature, there has been numerical work suggesting that condition (1) may not be true 7 for the MR state. One of the goals of our work is to perform more precise numerical calculations to determine whether this numerical conclusion holds up to more careful scrutiny. Condition (2) As qp's are moved apart from each other, the unitary transformation that results from adiabatically dragging one qp around another must converge to its...