We investigate the splitting probability of a monitored continuous-time quantum walk with two targets and show that, in stark contrast to a classical random walk, it exhibits a nonanalytic, phase-transition-like behavior controlled by the sampling time at the targets. For large systems and sampling times smaller than a critical value τ_{c}=2π/ΔE, where ΔE is the energy bandwidth, the splitting probability is universal and equal to 1/2, independent of the initial condition and the sampling time. Above the critical sampling time, a nonuniversal regime emerges in which the splitting probability deviates from 1/2 and develops a fluctuating pattern of pronounced peaks and dips dependent on both the sampling time and the initial condition. These results follow from a nontrivial mapping of the splitting problem onto a pair of single-target detection problems enabled by the superposition principle.
We investigate the splitting probability of a monitored continuous-time quantum walk with two targets and show that, in stark contrast to a classical random walk, it exhibits a nonanalytic, phase-transition-like behavior controlled by the sampling time at the targets. For large systems and sampling times smaller than a critical value τ_{c}=2π/ΔE, where ΔE is the energy bandwidth, the splitting probability is universal and equal to 1/2, independent of the initial condition and the sampling time. Above the critical sampling time, a nonuniversal regime emerges in which the splitting probability deviates from 1/2 and develops a fluctuating pattern of pronounced peaks and dips dependent on both the sampling time and the initial condition. These results follow from a nontrivial mapping of the splitting problem onto a pair of single-target detection problems enabled by the superposition principle.
We consider a continuous-time quantum walk stroboscopically subjected to projective measurements on two distinct targets with sampling time period τ. We construct a general renewal framework for the first-detection amplitude at each target, from which we obtain the associated splitting probabilities. For Hamiltonians with reflection symmetry, we further show that each dual-target first-detection amplitude can be viewed as an interference effect between a pair of single-target first-detection amplitudes. This formulation yields exact expressions for the splitting probabilities and gives special sampling times at which they undergo pointwise discontinuous jumps due to a form of ergodicity breaking associated with dark states. As an illustrative example, we apply the framework to a one-dimensional tight-binding lattice chain with nearest-neighbor hopping, and study the splitting probabilities as functions of the initial position and sampling time. Below a critical sampling time, 0<τ<τc where τc=2π/ΔE with ΔE being the energy bandwidth, the splitting probabilities for large system sizes become equal to 1/2, independent of the initial position, as long as this position does not coincide with target sites. Above the critical sampling time, τ≥τc, the splitting probabilities deviate from 1/2 and develop oscillations as a function of initial position. The wavenumber associated with these oscillations is Q=2sin−1(rτc/τ), where r is a positive integer. We also explore the effects of relative phase and spatial separation between different components of a coherent initial state on the splitting probabilities. Finally, we study how a finite number of measurements modifies the splitting probabilities and broadens the resonances at special sampling times.
In a quantum walk, the first-detection return probability (FDRP) characterizes salient features, determining whether the quantum walk is transient or recurrent. We study the FDRP of quantum walks on a chain where the initial site is stroboscopically monitored by a detector and the walker performs long-range hopping between sites. We assume that the hopping strength decays with the distance d as d−α and α≥0 and show that the power-law exponent α critically determines the behavior of the FDRP. The value α=1 separates recurrent (α<1) from transient (α>1) quantum walks through a continuous phase transition in the total detection probability. For α<1, strong long-range hopping induces localization, resulting in unit total detection probability. Instead, for α>1 the long-range walk is transient and the return probability decays algebraically as a function of time as t−β. The associated decay exponent β features nonanalytic points as a function of α. Such singularities are not exclusively determined by the low-energy spectrum, but are caused by the interference between infrared and ultraviolet energy modes induced by projective measurements, signalling the emergence of critical behavior intrinsic to the non-unitary dynamics. These dynamics are solely controlled by tuning the long-range exponent α and can thus be experimentally probed in atomic and molecular systems.
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.
Even after decades of research, the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time τ, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We illustrate our results with tight-binding quantum walk models. We examine a closed system, i.e., a ring, and reveal the intricate influence of the sampling time τ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals, and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases, the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one-dimensional quantum walk, the probability of first detection decays like (time)^{(-3)} with superimposed oscillations, with exceptional behavior when the sampling period τ times the tunneling rate γ is a multiple of π/2. The amplitude of the power-law decay is suppressed as τ→0 due to the Zeno effect. Our work, an extended version of our previously published paper, predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like (time)^{-3/2}, as elucidated by Schrödinger in 1915.
We investigate the genuinely quantum features of continuous-time quantum walks by combining a single-time and a multi-time quantifier of nonclassicality. On the one hand, we consider the quantum-classical dynamical distance DQC(t), which measures the departure of the time-evolved quantum state of a continuous-time quantum walk from the classical state of a random walk on the same graph. On the other, we analyse the joint probability distributions associated with sequential measurements of the walker's position, assessing their violation of the classical Kolmogorov consistency conditions via a dedicated quantifier Kˉ(t). We demonstrate a quadratic short-time scaling of Kˉ(t), which differs from the known linear scaling of DQC(t), but, as the latter, is fully determined by the degree of the initially occupied node and is independent of the global graph topology. At longer times, instead, Kˉ(t) exhibits a pronounced topology-driven behavior: it is strongly suppressed on complete graphs while remaining finite and oscillatory on cycles, in contrast with the almost topology-independent asymptotics of DQC(t). We then extend the analysis to Markovian open-system dynamics, focusing on dephasing in the position basis (Haken-Strobl model) and in the energy basis (intrinsic decoherence). Site dephasing drives both quantifiers to zero, with the decay of Kˉ(t) controlled by the spectral gap of the corresponding Lindblad generator. By contrast, energy-basis dephasing preserves a finite asymptotic value of Kˉ(t), depending on the overlap structure of the Laplacian eigenspaces with the site basis.
Continuous-time quantum walks with long-range steps ${R}^{\ensuremath{-}\ensuremath{\gamma}}$ (R being the distance between sites) on a discrete line behave in similar ways for all $\ensuremath{\gamma}\ensuremath{\ge}2$. This is in contrast to classical random walks, which for $\ensuremath{\gamma}>3$ belong to a different universality class than for $\ensuremath{\gamma}\ensuremath{\le}3$. We show that the average probabilities to be at the initial site after time t as well as the mean square displacements are of the same functional form for quantum walks with $\ensuremath{\gamma}=2$, 4, and with nearest neighbor steps. We interpolate this result to arbitrary $\ensuremath{\gamma}\ensuremath{\ge}2$.