Motivated by the underlying desire to identify novel, physically reasonable resource states for measurement-based quantum computation (MBQC), this thesis explores two seemingly unrelated topics in some detail. The first is a study of the circumstances under which multiqubit quantum states that are equivalent to cluster states of the same size under stochastic local operations and classical communication (SLOCC) are either deterministic or probabilistic resource states, with the aim of identifying new resource states that are related to, but non-trivially different from, the cluster states.The second is an analysis of the properties of the ground state of a potentially physically realisable coupled-cavity quantum electrodynamics model called the Jaynes-Cummings-Hubbard (JCH) model. The hope is that the ground state of the model can in fact serve as a universal resource for MBQC.
In the first study, I identify two classes of 1D states in the
SLOCC-equivalence class of 1D cluster states that constitute resources for random-length single-qubit rotations, in one case quasi-deterministically (N-U-N states) and in another probabilistically (B-U-B states). In contrast to the cluster states, the N-U-N states exhibit spin correlation functions that decay exponentially with distance, while the B-U-B states can be arbitrarily locally pure. I also show that a two-dimensional square N-U-N lattice is a universal resource for quasi-deterministic measurement-based quantum computation, and that cubic B-U-B states can be locally converted to 2D universal resource states.
In the second study, the Density Matrix Renormalization Group (DMRG) algorithm is used to characterize the ground states of the 1D JCH model in the regime of low photon densities, and compare it to the 1D ground state of the Bose-Hubbard (BH) model. Numerical results indicate that a Tonks-Girardeau regime, in which the photons are strongly fermionized, appears between the Mott-insulating and superfluid phases as a function of the intercavity coupling.
The final chapter of the thesis outlines the initial progress that I have made in determining whether the 1D JCH ground state can serve as a resource for universal single-qubit rotations in the MBQC picture, as well as directions for future investigation.