Quantum Holonomy for Many-Body Systems and Quantum Computation

Abstract: The research of this Ph. D. thesis is in the field of Quantum Computation and QuantumInformation. A key problem in this field is the fragile nature of quantum states. Thisbecomes increasingly acute when the number of quantum bits (qubits) grows in order toperform large quantum computations. It has been proposed that geometric (Berry) phasesmay be a useful tool to overcome this problem, because of the inherent robustness of suchphases to random noise. In the thesis we investigate geometric phases and quantumholonomies (matrix-valued geometric phases) in many-body quantum systems, and elucidatethe relationship between these phases and the quantum correlations present in the systems.An overall goal of the project is to assess the feasibility of using geometric phases andquantum holonomies to build robust quantum gates, and investigate their behavior when thesize of a quantum system grows, thereby gaining insights into large-scale quantumcomputation.In a first project we study the Uhlmann holonomy of quantum states for hydrogen-likeatoms. We try to get into a physical interpretation of this geometric concept by analyzing itsrelation with quantum correlations in the system, as well as by comparing it with differenttypes of geometric phases such as the standard pure state geometric phase, Wilczek-Zeeholonomy, Lévay geometric phase and mixed-state geometric phases. In a second project weestablish a unifying connection between the geometric phase and the geometric measure ofentanglement in a generic many-body system, which provides a universal approach to thestudy of quantum critical phenomena. This approach can be tested experimentally in aninterferometry setup, where the geometric measure of entanglement yields the visibility ofthe interference fringes, whereas the geometric phase describes the phase shifts. In a thirdproject we propose a scheme to implement universal non-adiabatic holonomic quantumgates, which can be realized in novel nano-engineered systems such as quantum dots,molecular magnets, optical lattices and topological insulators. In a fourth project we proposean experimentally feasible approach based on “orange slice” shaped paths to realize non-Abelian geometric phases, which can be used particularly for geometric manipulation ofqubits. Finally, we provide a physical setting for realizing non-Abelian off-diagonalgeometric phases. The proposed setting can be implemented in a cyclic chain of four qubitswith controllable nearest-neighbor interactions. Our proposal seems to be within reach invarious nano-engineered systems and therefore opens up for first experimental test of thenon-Abelian off-diagonal geometric phase.