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arXiv:2402.15843 · Entropy

First Hitting Times on a Quantum Computer: Tracking vs. Local Monitoring, Topological Effects, and Dark States

Q. Q. Wang, Silin Ren, Ruoyu Yin, K. Ziegler, Eli Barkai, Sabine Tornow

Published Oct 16, 2024• 15 citations
arXivHTMLPDF

Abstract

We investigate a quantum walk on a ring represented by a directed triangle graph with complex edge weights and monitored at a constant rate until the quantum walker is detected. To this end, the first hitting time statistics are recorded using unitary dynamics interspersed stroboscopically by measurements, which are implemented on IBM quantum computers with a midcircuit readout option. Unlike classical hitting times, the statistical aspect of the problem depends on the way we construct the measured path, an effect that we quantify experimentally. First, we experimentally verify the theoretical prediction that the mean return time to a target state is quantized, with abrupt discontinuities found for specific sampling times and other control parameters, which has a well-known topological interpretation. Second, depending on the initial state, system parameters, and measurement protocol, the detection probability can be less than one or even zero, which is related to dark-state physics. Both return-time quantization and the appearance of the dark states are related to degeneracies in the eigenvalues of the unitary time evolution operator. We conclude that, for the IBM quantum computer under study, the first hitting times of monitored quantum walks are resilient to noise. However, a finite number of measurements leads to broadening effects, which modify the topological quantization and chiral effects of the asymptotic theory with an infinite number of measurements.

DOI
10.3390/e26100869

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We investigate a quantum walk on a ring represented by a directed triangle graph with complex edge weights and monitored at a constant rate until the quantum walker is detected. To this end, the first hitting time statistics are recorded using unitary dynamics interspersed stroboscopically by measurements, which are implemented on IBM quantum computers with a midcircuit readout option. Unlike classical hitting times, the statistical aspect of the problem depends on the way we construct the measured path, an effect that we quantify experimentally. First, we experimentally verify the theoretical prediction that the mean return time to a target state is quantized, with abrupt discontinuities found for specific sampling times and other control parameters, which has a well-known topological interpretation. Second, depending on the initial state, system parameters, and measurement protocol, the detection probability can be less than one or even zero, which is related to dark-state physics. Both return-time quantization and the appearance of the dark states are related to degeneracies in the eigenvalues of the unitary time evolution operator. We conclude that, for the IBM quantum computer under study, the first hitting times of monitored quantum walks are resilient to noise. However, a finite number of measurements leads to broadening effects, which modify the topological quantization and chiral effects of the asymptotic theory with an infinite number of measurements.
We study the first detected recurrence time problem of continuous-time quantum walks on graphs. While previous works have employed projective measurements to determine the first return time, we implement a protocol based on weak measurements on a dilated system, enabling minimally invasive monitoring throughout the evolution. To achieve this, we implement a weak measurement protocol theoretically and complement it with both numerical simulations and investigations on an IBM quantum computer. Despite the implementation of a generalized measurement, the proposed protocol provides a description purely within the Hilbert space of the quantum system. Our results reveal that for rank-one monitored, pure initial states, the first hitting timescales inversely with the coupling parameter between the ancilla and the quantum system.
We study a quantum walk of a single particle that is subject to stroboscopic projective measurements on a graph with two sites. This two-level system is the minimal model of a measurement-induced quantum walk. The mean first detected transition and return time are computed on an IBM quantum computer as a function of the hopping matrix element between the sites and the on-site potential. The experimentally monitored quantum walk reveals the theoretically predicted behavior, such as the quantization of the first detected return time and the strong increase of the mean first detected transition time near degenerate points, with high accuracy.
We investigate a tight-binding quantum walk on a graph. Repeated stroboscopic measurements of the position of the particle yield a measured "trajectory," and a combination of classical and quantum mechanical properties for the walk are observed. We explore the effects of the measurements on the spreading of the packet on a one-dimensional line, showing that except for the Zeno limit, the system converges to Gaussian statistics similarly to a classical random walk. A large deviation analysis and an Edgeworth expansion yield quantum corrections to this normal behavior. We then explore the first passage time to a target state using a generating function method, yielding properties like the quantization of the mean first return time. In particular, we study the effects of certain sampling rates that cause remarkable changes in the behavior in the system, such as divergence of the mean detection time in finite systems and decomposition of the phase space into mutually exclusive regions, an effect that mimics ergodicity breaking, whose origin here is the destructive interference in quantum mechanics. For a quantum walk on a line, we show that in our system the first detection probability decays classically like (time)^{-3/2}. This is dramatically different compared to local measurements, which yield a decay rate of (time)^{-3}, indicating that the exponents of the first passage time depend on the type of measurements used.
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.
The problem of quantum first detection time has been extensively investigated in recent years. Employing the stroboscopic measurement protocol, we consider such a monitored quantum walk on sequentially or periodically perturbed rings and focus on the statistics of first detected return time, namely, the time it takes a particle to return to the initial state for the first time. Using time-independent perturbation theory, we obtain the general form of the eigenvalues and eigenvectors of the Hamiltonian. For the case of a sequentially perturbed ring system, we find steplike behaviors of ${\ensuremath{\sum}}_{n=1}^{\mathcal{N}}n{F}_{n}$ ($\ensuremath{\rightarrow}\ensuremath{\langle}n\ensuremath{\rangle}$ as $\mathcal{N}\ensuremath{\rightarrow}\ensuremath{\infty}$) when N\mathcal{N}N increases, with two plateaus corresponding to integers, where Fn{F}_{n}Fn​ is the first detected return probability at the nthn\mathrm{th}nth detection attempt. Meanwhile, if the initial condition preserves the reflection symmetry, the mean return time is the same as the unperturbed system. For the periodically perturbed system, similar results can also appear in the case where the symmetry is preserved; however, the size of the ring, the interval between adjacent perturbations, and the initial position may change the mean return time in most cases. In addition, we find that the decay rate of the first detection probability Fn{F}_{n}Fn​ decreases with the increase in perturbation amplitude. More profoundly, the symmetry-preserving setup of the initial conditions leads to the coincidence of Fn{F}_{n}Fn​. The symmetry of the physical systems under investigation is deeply reflected in the quantum detection time statistics.
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.