Entanglement and Quantum Computation from a Geometric and Topological Perspective

Sammanfattning: In this thesis we investigate geometric and topological structures in the context of entanglement and quantum computation.A parallel transport condition is introduced in the context of Franson interferometry based on the maximization of two-particle coincidence intensity. The dependence on correlations is investigated and it is found that the holonomy group is in general non-Abelian, but Abelian for uncorrelated systems. It is found that this framework contains a parallel transport condition developed by Levay in the case of two-qubit systems undergoing local SU(2) evolutions.Global phase factors of topological origin, resulting from cyclic local SU(2) evolution, called topological phases, are investigated in the context of multi-qubit systems. These phases originate from the topological structure of the local SU(2)-orbits and are an attribute of most entangled multi-qubit systems. The relation between topological phases and SLOCC-invariant polynomials is discussed. A general method to find the values of the topological phases in an n-qubit system is described.A non-adiabatic generalization of holonomic quantum computation is developed in which high-speed universal quantum gates can be realized by using non-Abelian geometric phases. It is shown how a set of non-adiabatic holonomic one- and two-qubit gates can be implemented by utilizing transitions in a generic three-level ? configuration. The robustness of the proposed scheme to different sources of error is investigated through numerical simulation. It is found that the gates can be made robust to a variety of errors if the operation time of the gate can be made sufficiently short. This scheme opens up for universal holonomic quantum computation on qubits characterized by short coherence times.