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We introduce a novel time-energy uncertainty relationship within the contextof restarts in monitored quantum dynamics. Initially, we investigate theconcept of ``first hitting time'' in quantum systems using an IBM quantumcomputer and a three-site ring graph as our starting point. Previous studieshave established that the mean recurrence time, which represents the time takento return to the initial state, is quantized as an integer multiple of thesampling time, displaying pointwise discontinuous transitions at resonances.Our findings demonstrate that, the natural utilization of the restart mechanismin laboratory experiments, driven by finite data collection time spans, leadsto a broadening effect on the transitions of the mean recurrence time. Ournewly proposed uncertainty relation captures the underlying essence of thesephenomena, by connecting the broadening of the mean hitting time nearresonances, to the intrinsic energies of the quantum system and to thefluctuations of recurrence time. This work not only contributes to ourunderstanding of fundamental aspects related to quantum measurements anddynamics, but also offers practical insights for the design of efficientquantum algorithms with mid-circuit measurements.

Restart uncertainty relation for monitored quantum dynamics