We introduce a time-energy uncertainty relation within the context of restarts in monitored quantum dynamics. Previous studies have established that the mean recurrence time, which represents the time taken to return to the initial state, is quantized as an integer multiple of the sampling time, displaying pointwise discontinuous transitions at resonances. Our findings demonstrate that the natural utilization of the restart mechanism in laboratory experiments, driven by finite data collection time spans, leads to a broadening effect on the transitions of the mean recurrence time. Our proposed uncertainty relation captures the underlying essence of these phenomena, by connecting the broadening of the mean hitting time near resonances, to the intrinsic energies of the quantum system and to the fluctuations of recurrence time. Our uncertainty relation has also been validated through remote experiments conducted on an International Business Machines Corporation (IBM) quantum computer. This work not only contributes to our understanding of fundamental aspects related to quantum measurements and dynamics, but also offers practical insights for the design of efficient quantum algorithms with mid-circuit measurements.
We introduce a time-energy uncertainty relation within the context of restarts in monitored quantum dynamics. Previous studies have established that the mean recurrence time, which represents the time taken to return to the initial state, is quantized as an integer multiple of the sampling time, displaying pointwise discontinuous transitions at resonances. Our findings demonstrate that the natural utilization of the restart mechanism in laboratory experiments, driven by finite data collection time spans, leads to a broadening effect on the transitions of the mean recurrence time. Our proposed uncertainty relation captures the underlying essence of these phenomena, by connecting the broadening of the mean hitting time near resonances, to the intrinsic energies of the quantum system and to the fluctuations of recurrence time. Our uncertainty relation has also been validated through remote experiments conducted on an International Business Machines Corporation (IBM) quantum computer. This work not only contributes to our understanding of fundamental aspects related to quantum measurements and dynamics, but also offers practical insights for the design of efficient quantum algorithms with mid-circuit measurements.
The recurrence time is the time a process first returns to its initial state. Using quantum walks on a graph, the recurrence time is defined through the stroboscopic monitoring of the arrival of the particle to a node of the system. When the time interval between repeated measurements is tuned in such a way that the eigenvalues of the unitary become degenerate, the mean recurrence time exhibits resonances. These resonances imply faster mean recurrence times, which were recorded on quantum computers. The resonance broadening is captured by a restart uncertainty relation [Yin et al., Proc. Natl. Acad. Sci. U.S.A. 122, e2402912121 (2025)]. To ensure a comprehensive analysis, we extend our investigation to include the impact of system size on the widened resonances, showing how the connectivity and energy spectrum structure of a system influence the restart uncertainty relation. Breaking the symmetry of the system, for example time-reversal symmetry breaking with a magnetic flux applied to a ring, removes the degeneracy of the eigenvalues of the unitary, hence modifying the mean recurrence time and the widening of the transitions, and this effect is studied in detail. The width of the resonances studied here is related to the finite time resolution of relevant experiments on quantum computers and to the restart paradigm.
We study non-equilibrium steady states and recurrence times in noisy, stroboscopically monitored qubit systems using complete measurements. In the noiseless limit, recurrence times are integer-quantized, with dips to lower integers when sampling approaches revival conditions associated with ergodicity breaking. Using an IBM quantum platform, we find that quantization is robust when sampling far from revivals, but breaks down dramatically near revivals: even weak noise produces large deviations and can invert the expected dips into pronounced peaks. To explain this behavior, we formulate a statistical-physics model of monitored noisy circuits in which monitoring drives an effective infinite-temperature steady state while thermal-like relaxation competes to favor a low-temperature limit. We show that the sampling time tunes a crossover between these regimes, near revivals stabilizing low-temperature behavior, and far from revivals restoring infinite-temperature behavior -- with noise strength and detuning acting as coupled small parameters near resonance.
Recurrence time quantifies the duration required for a physical system to return to its initial state, playing a pivotal role in understanding the predictability of complex systems. In quantum systems with subspace measurements, recurrence times are governed by Anandan-Aharonov phases, yielding fractionally quantized recurrence times. However, the fractional quantization phenomenon in interacting quantum systems remains unexplored. Here, we address this gap by establishing universal lower and upper bounds for recurrence times in interacting many-body spin systems. Notably, we investigate scenarios where these bounds are approached, shedding light on the speed of quantum processes under monitoring. In specific cases, our findings reveal that the complex many-body system can be effectively mapped onto a dynamical system with a single quasi-particle, leading to integer-quantized recurrence times. Our work demonstrates a valuable link between recurrence times and the number of dark states in the system, thus providing a deeper understanding of the intricate interplay between Hilbert-space fragmentation, ergodicity breaking, measurements, and interaction effects. Finally, our findings have been implemented on an IBM quantum computer, revealing resonances and fractional quantization in agreement with theoretical predictions. This demonstrates the resilience of nonequilibrium topological fractional quantization to noise and highlights its potential use for benchmarking quantum devices and probing dark states.
Repeatedly monitored quantum walks with a rate 1/τ yield discrete-time trajectories which are inherently random. With these paths the first-hitting time with sharp restart is studied. We find an instability in the optimal mean hitting time, which is not found in the corresponding classical random-walk process. This instability implies that a small change in parameters can lead to a rather large change of the optimal restart time. We show that the optimal restart time versus τ, as a control parameter, exhibits sets of staircases and plunges. The plunges, are due to the mentioned instability, which in turn is related to the quantum oscillations of the first-hitting time probability, in the absence of restarts. Furthermore, we prove that there are only two patterns of staircase structures, dependent on the parity of the distance between the target and the source in units of lattice constant. The global minimum of the hitting time is controlled not only by the restart time, as in classical problems, but also by the sampling time τ. We provide numerical evidence that this global minimum occurs for the τ minimizing the mean hitting time, given restarts taking place after each measurement. Last, we numerically show that the instability found in this work is relatively robust against stochastic perturbations in the sampling time τ.
Classical first-passage times under restart are used in a wide variety of models, yet the quantum version of the problem still misses key concepts. We study the quantum hitting time with restart using a monitored quantum walk. The restart strategy eliminates the problem of dark states, i.e., cases where the particle evades detection, while maintaining the ballistic propagation which is important for a fast search. We find profound effects of quantum oscillations on the restart problem, namely, a type of instability of the mean detection time, and optimal restart times that form staircases, with sudden drops as the rate of sampling is modified. In the absence of restart and in the Zeno limit, the detection of the walker is not possible, and we examine how restart overcomes this well-known problem, showing that the optimal restart time becomes insensitive to the sampling period.
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.