Geometric Control For Analysing the Quantum Speed Limit and the Physical Limitations of Computers

This thesis studies the role of Finsler geometry in quantum time optimal control of systems with constrained control field power and other constraints. The systems considered are all finite dimensional systems with pure states. A Finsler metric is constructed such that its geodesics are the time optimal trajectories for the quantum time evolution operator on the special unitary group. This metric is shown to be right invariant. The geodesic equation, in the form of an Euler-Poincar\'{e} equation is found. It is also shown that the geodesic lengths of this same metric equal the optimal times for implementing any desired quantum gate. In a special case, where all are control fields are equally constrained, the desired geodesics are found in closed form. The results obtained are discussed in the general context of natural computation.