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arXiv:2506.21168 · Physical Review A

Quantum walks: First hitting times with weak measurements

Tim Heine, Eli Barkai, K. Ziegler, Sabine Tornow

Published Apr 10, 2026• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/j2yb-fmw1

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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.
Measurements can be used to monitor the evolution of quantum systems and can give rise to quantized return statistics. It is known that the mean return time is quantized for strong monitoring through the winding number of the monitored quantum state. We discuss that under coherent weak monitoring, implemented via ancilla coupling, the mean return time of a quantum walk obeys a scaling relation with respect to the measurement strength. An analog scaling relation was previously found for random-time monitoring, indicating that weak and random-time monitoring have similar effects. We discuss how weak monitoring via ancilla coupling is linked to the unitary evolution, and how this connection can be controlled by a convergent perturbation theory.
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.
The topics presented in this thesis have continuous quantum measurements and quantum walks at their core. The first topic being discussed centers around simulating a generalize measurement with a finite number of outcomes using a continuous measurement process with a continuous measurement history. We provide conditions under which it is possible to prove that such a process exists and that at long times it simulates faithfully the generalized measurement. We give the stochastic equations governing the feedback between the measurement history and the instantaneous weak measurements. The second topic examines a definition of hitting time for continuous-time quantum walks. A crucial component for such a definition is the use of weak measurements. Several methods using alternative but equivalent definitions of weak, continuous measurements are employed to derive a formula for the hitting time. The behavior of the thus defined hitting time is studied subsequently, in general and for specific graphs. The last topic explores continuous-time quantum walks on graphs with infinite tails. The equations for propagating and bound states are derived and the S-matrix is defined. Their properties, such as orthogonality of the propagating and bound states, unitarity of the S-matrix, are discussed. Formulas for the S-matrix under operations of cutting, adding or connecting tails are derived.
Measurements can be used to monitor the evolution of quantum systems and may lead to a universally quantized time statistics. It is known that the mean return time is quantized for strong and indirect monitoring through the winding number of the return amplitude in a one-dimensional space. Here we discuss that under multi-channel strong or indirect monitoring, where the latter is achieved through ancilla coupling, the mean return time of a quantum walk in the projected subspace is also quantized. This reflects a universal time quantization for a higher dimensional evolution.
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 consider the first detection problem for a one-dimensional quantum walk with repeated local measurements. Employing the stroboscopic projective measurement protocol and the renewal equation, we study the effect of tunneling on the detection time. Specifically, we study the continuous-time quantum walk on an infinite tight-binding lattice for two typical situations with physical reality. The first is the case of a quantum walk in the absence of tunneling with a Gaussian initial state. The second is the case where a barrier is added to the system. It is shown that the transition of the decay behavior of the first detection probability can be observed by modifying the initial condition, and in the presence of a tunneling barrier, the particle can be detected earlier than the impurity-free lattice. This suggests that the evolution of the walker is expedited when it tunnels through the barrier under repeated measurement. The first detection tunneling time is introduced to investigate the tunneling time of the quantum walk. In addition, we analyze the critical transitive point by deriving an asymptotic formula.