In this article we explore a modification in the problem of controlling the rotation of a two level quantum system from an initial state to a final state in minimum time. Specifically we consider the case where the qubit is being weakly monitored -- albeit with an assumption that both the measurement strength as well as the angular velocity are assumed to be control signals. This modification alters the dynamics significantly and enables the exploitation of the measurement backaction to assist in achieving the control objective. The proposed method yields a significant speedup in achieving the desired state transfer compared to previous approaches. These results are demonstrated via numerical solutions for an example problem on a single qubit.
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