Topological quantum computation provides an architecture for encoding quantum information in such a way as to be theoretically robust to local noise. Logical qubits are encoded in topological degrees of freedom typically in spin-lattice models. Excitations in the spin-lattice models manifest as anyons, generalisations of bosons and fermions that exist in two dimensions. Proposals for experimental realisation of these topological systems have previously relied on perfect anyon creation in a time independent-manner or in the case of time-dependent proposals have primarily relied on adiabatic, therefore slow, dynamics. The aim of this thesis is to use quantum control to create anyons and encode logical qubits in topological systems without the requirement for adiabaticity and long timescales.
First, we demonstrate the creation of abelian anyons using time-dependent controls in the toric code model, a system that is useful as a quantum memory. We show that this may be done within arbitrarily short timescales at the expense of larger magnitude control pulses. Additionally, we investigate the robustness of our protocol in the face of theoretical errors in anyon creation.
Secondly, we investigate the creation of non-abelian anyons in the Kitaev honeycomb model. By fermionising a time-dependent version of the model we demonstrate how optimal control theory can allow for anyon creation in faster-than-adiabatic time. Moreover, we show that the particular method we develop to achieve this scales only linearly in the number of spins in the lattice.
Thirdly, we investigate defect creation in the surface code, a generalisation of the toric code that does not require periodic boundary conditions. Optimal quantum control is used to show how defects may be created faster than with the typical adiabatic procedures. Additionally, a method using mapping of dynamical Lie algebras is used to demonstrate that optimal control techniques may be extended to operations whose dynamics require solving in a large Hilbert space.