  In the geometry of quantum evolutions, a geodesic path is viewed as a path of
minimal statistical length connecting two pure quantum states along which the
maximal number of statistically distinguishable states is minimum. In this
paper, we present an explicit geodesic analysis of the dynamical trajectories
that emerge from the quantum evolution of a single-qubit quantum state. The
evolution is governed by an Hermitian Hamiltonian operator that achieves the
fastest possible unitary evolution between given initial and final pure states.
Furthermore, in addition to viewing geodesics in ray space as paths of minimal
length, we also verify the geodesicity of paths in terms of unit geometric
efficiency and vanishing geometric phase. Finally, based on our analysis, we
briefly address the main hurdles in moving to the geometry of quantum
evolutions for open quantum systems in mixed quantum states.
