In this paper we extend the concept of persistence, well defined for classical stochastic dynamics, to the context of quantum dynamics. We demonstrate the idea via quantum random walk and a successive measurement scheme, where persistence is defined as the time during which a given site remains unvisited by the walker. We also investigated the behavior of related quantities, e.g., the first-passage time and the succession probability (newly defined), etc. The study reveals power-law scaling behavior of these quantities with new exponents. Comparable features of the classical and the quantum walks are discussed.
We study a discrete-time quantum walk in presence of a detector at
xD initially. The detector here is repeatedly removed after a span of
tR, the removal time, and reinserted at random locations. Two relocation rules are considered here: In Model~1, the detector is reinserted at any site beyond
xD, while in Model~2, reinsertion is done within a restricted window around the position of the detector at that time. Both variants behave like Semi Infinite Walk (SIW) for large
tR, where the detector behaves effectively as a fixed boundary. However, in the rapid-relocation regime, i.e., when
tR is small, the behaviours are different. Model~1 permits greater spreading due to unrestricted reinsertion, which is different from Model~2. The time evolution of occupation probability ratio of our walker to that of an infinite walker at
xD, i.e.,
f(xD,t)/f∞(xD,t), initially show the feature of a SIW upto
t=tR, then show some oscillatory behaviour and finally reach a saturation value for both the models. The ratio enhancing under certain conditions of
xD and
tR, is a purely quantum mechanical effect. The saturation ratio shows a crossover behavior below and above a removal time
tR∗. At sites
x=xD the occupation probablity ratios at a certain time reveals that for small
tR, the behaviours of the two models are drastically different from each other, as well as from Semi Infinite Walk (SIW), Quenched Quantum Walk (QQW) and Moving Detector Quantum Walk (MDQW). The correlation ratios of the two models with that of Infinite Walk (IW) show interesting time dependence for sites to the left or right of the initial detector position
xD.
We analyze the asymptotic scaling of persistence of unvisited sites for quantum walks on a line. In contrast to the classical random walk, there is no connection between the behavior of persistence and the scaling of variance. In particular, we find that for a two-state quantum walk persistence follows an inverse power law where the exponent is determined solely by the coin parameter. Moreover, for a one-parameter family of three-state quantum walks containing the Grover walk, the scaling of persistence is given by two contributions. The first is the inverse power law. The second contribution to the asymptotic behavior of persistence is an exponential decay coming from the trapping nature of the studied family of quantum walks. In contrast to the two-state walks, both the exponent of the inverse power-law and the decay constant of the exponential decay depend also on the initial coin state and its coherence. Hence, one can achieve various regimes of persistence by altering the initial condition, ranging from purely exponential decay to purely inverse power-law behavior.
Quantum walks exhibit many unique characteristics compared to classical random walks. In the classical setting, self-avoiding random walks have been studied as a variation on the usual classical random walk. Classical self-avoiding random walks have found numerous applications, most notably in the modeling of protein folding. We consider the analogous problem in the quantum setting. We complement a quantum walk with a memory register that records where the walker has previously resided. The walker is then able to avoid returning back to previously visited sites. We parameterise the strength of the memory recording and the strength of the memory back-action on the walker's motion, and investigate their effect on the dynamics of the walk. We find that by manipulating these parameters the walk can be made to reproduce ideal quantum or classical random walk statistics, or a plethora of more elaborate diffusive phenomena. In some parameter regimes we observe a close correspondence between classical self-avoiding random walks and the quantum self-avoiding walk.
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 $\ensuremath{\gamma}\ensuremath{\ge}2$.
We present a detailed introduction to the discrete-time quantum walk problem, in close analogy with the classical ordinary and persistent random walk. This approach facilitates a uniform application of the renormalization group that highlights similarities and differences between the classical and the quantum walk problem. Specifically, we discuss the renormalization group treatment for the mean-square displacement of a walker starting from a single site on the 1d-line for ordinary and persistent random walks and the quantum walk. We outline the significance of universality for quantum walks and the control this might provide for quantum algorithms. We use our RG method to verify that all 2-state quantum walks on the 1d-line are in the same universality class.