Fidelity of geometric and holonomic quantum gates for spin systems

Geometric and holonomic quantum gates perform transformations that only dependon the geometry of a loop covered by the parameters controlling the gate. Thesegates require adiabatic time evolution, which is achieved in the limit when the looptakes infinite time to complete. However, it is of interest to also know thetransformation properties of the gates for finite run times. It has been shown [Phys.Rev. A 73, 022327 (2006)] that some holonomic gates for a trapped ion system showrevival structures, i.e., for some finite run time the gate performs the sametransformation as it does in the adiabatic limit. The purpose of this thesis is to investigate if similar revival structures are shown alsofor geometric and holonomic quantum gates for spin systems. To study geometricquantum gates an NMR setup for spin-1/2 particles is used, while an NQR setup forspin-3/2 particles is used to study holonomic quantum gates. Furthermore, for thegeometric quantum gates the impact of some open system effects are examined byusing the quantum jump approach. The non-adiabatic time evolution operators of thesystems are calculated and compared to the corresponding adiabatic time evolutionoperators by computing their operator fidelity. The operator fidelity ranges between0 and 1, where 1 means that the gates are identical up to an unimportant phasefactor. All gates show an oscillating dependency on the run time, and some Abeliangates even show true revivals, i.e., the operator fidelity reaches 1