Adiabatic transition from the cluster state to the surface code

This project studies two different quantum phases of matter and how to design systems that adiabatically connect one phase with another. One phase corresponds to a cluster state which is a resource state for measurement-based quantum computation, and the other is a surface code which is a robust way to store quantum memory. Both phases are ground states of strongly correlated two-dimensional lattices of quantum systems, either two-level systems (qubits, fermions) or infinite-dimensional, continuous-variable (CV) systems (quantum modes, bosons), and both phases are gapped. However. the surface code has a special kind of non-local order, termed topological order, while the cluster phase does not. A key advantage of the cluster phase is that it can be relatively easily prepared in experiment using a constant depth circuit acting on an initially unentangled state. The surface code, in contrast, requires a number of preparation steps that scales with the system size; a consequence of the long range topological order in this phase. Remarkably, it has been shown that the surface code can be prepared from the cluster phase simply by performing a pattern of commuting single site measurements on the lattice. However, for any outcome of measurements, it is necessary to perform a set of corrections to the state such that the total preparation time is still extensive. The focus of this thesis is how to smoothly perform the entire preparation procedure for the surface code by deforming a Hamiltonian which encodes the state in the ground state. This avoids measurement altogether and moreover has the advantage that for CV systems the Hamiltonian involves only two-body near-neighbour interactions rather than the four-body interactions that are required in a spin encoding. In this thesis, we study a smooth, adiabatic transition from the cluster state to the surface code. We do this in a series of steps: We first consider the adiabatic evolution of a single qubit and a single qumode. We also calculate the iterative, discrete time step approximation of the continuous adiabatic evolution of states in the qubit case, and begin the work toward the adiabatic evolution of CV operators by first considering a single qumode transition Hamiltonian. Then we study the spectrum of au adiabatic transition from a bosonic CV cluster state Hamiltonian to a CV planar surface code Hamiltonian. In particular, we track the energy gap between the ground and first excited state during the transition.