Grovers 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, Grovers quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagins 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 Schrodingers equation are represented as vector fields. In view of optimal control, Grovers 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 Pontryagins Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.
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