  We use geometric concepts originally proposed by Anandan and Aharonov to show
that the Farhi-Gutmann time optimal analog quantum search evolution between two
orthogonal quantum states is characterized by unit efficiency dynamical
trajectories traced on a projective Hilbert space. In particular, we prove that
these optimal dynamical trajectories are the shortest geodesic paths joining
the initial and the final states of the quantum evolution. In addition, we
verify they describe minimum uncertainty evolutions specified by an uncertainty
inequality that is tighter than the ordinary time-energy uncertainty relation.
We also study the effects of deviations from the time optimality condition from
our proposed Riemannian geometric perspective. Furthermore, after pointing out
some physically intuitive aspects offered by our geometric approach to quantum
searching, we mention some practically relevant physical insights that could
emerge from the application of our geometric analysis to more realistic
time-dependent quantum search evolutions. Finally, we briefly discuss possible
extensions of our work to the geometric analysis of the efficiency of thermal
trajectories of relevance in quantum computing tasks.
