  Superoscillation is a counterintuitive phenomenon for its mathematical
feature of "faster-than-Fourier", which has allowed novel optical imaging
beyond the diffraction limit. Here, we provide a superoscillating quantum
control protocol realized by sequential selections in the framework of weak
measurement, which drives the apparatus (target) by repeatedly applying optimal
pre- and post-selections to the system (controller). Our protocol accelerates
the adiabatic transport of trapped ions and adiabatic quantum search algorithm
at a finite energy cost. We demonstrate the accuracy and robustness of the
protocol in the presence of decoherence and fluctuating noise and elucidate the
trade-off between fidelity and rounds of selections. Our findings provide
avenues for quantum state control and wave-packet manipulation using
superoscillation in quantum platforms such as trapped ions.
