  Grover's algorithm is one of the most famous algorithms which explicitly
demonstrates how the quantum nature can be utilized to accelerate the searching
process. In this work, Grover's quantum search problem is mapped to a
time-optimal control problem. Resorting to Pontryagin's Minimum Principle we
find that the time-optimal solution has the bang-singular-bang structure. This
structure can be derived naturally, without integrating the differential
equations, using the geometric control technique where Hamiltonians in the
Schr\"odinger's equation are represented as vector fields. In view of optimal
control, Grover's algorithm uses the bang-bang protocol to approximate the
optimal protocol with a minimized number of bang-to-bang switchings to reduce
the query complexity. Our work provides a concrete example how Pontryagin's
Minimum Principle is connected to quantum computation, and offers insight into
how a quantum algorithm can be designed.
