Ainsworth, Robert
(2014)
Optimising Qubit Designs for
Topological Quantum
Computation.
PhD thesis, National University of Ireland Maynooth.
Abstract
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 multiqubit systems) as a function of the number of particles composing
them. Specific results are given for the limits placed on ddimensional
qudits where logic gates are imposed by braiding anyons in 2+1 dimensions.
We also study qudits designed from ringshaped, anyonlike 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 nontrivially 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 nonAbelian 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.
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