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 $\ensuremath{\gamma}\ensuremath{\ge}2$.
Continuous-time quantum walks with long-range steps R\ensuremath−\ensuremathγ (R being the distance between sites) on a discrete line behave in similar ways for all \ensuremathγ\ensuremath≥2. This is in contrast to classical random walks, which for \ensuremathγ>3 belong to a different universality class than for \ensuremathγ\ensuremath≤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 \ensuremathγ=2, 4, and with nearest neighbor steps. We interpolate this result to arbitrary \ensuremathγ\ensuremath≥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.
A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at x=0 at time t=0, the persistence probability P(x,t) that a site at x has not been visited till time t has been calculated. P(x,t) behaves as (t/∣x∣−1)−α with α∼0.3 while the global fraction P(t)=∑xP(x,t)/2t of sites remaining unvisited at time t attains a constant value. F(x,t), the probability that the site at x is visited for the first time at t behaves as (t/∣x∣−1)−β/∣x∣ where β=1+α for t/∣x∣>>1,and F(t)=∑xF(x,t)/2t∼1/t. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.
Abstract. We study quantum transport on finite discrete structures and we model the process by means of continuous-time quantum walks. A direct and effective comparison between quantum and classical walks can be attained based on the average displacement of the walker as a function of time. Indeed, a fast growth of the average displacement can be advantageously exploited to build up efficient search algorithms. By means of analytical and numerical investigations, we show that the finiteness and the inhomogeneity of the substrate jointly weaken the quantum walk performance. We further highlight the interplay between the quantum-walk dynamics and the underlying topology by studying the temporal evolution of the transfer probability distribution and the lower bound of long time averages. PACS numbers: 05.60Gg, 71.35.-y, 05.60CdDynamics of continuous-time quantum walks in restricted geometries 2 1.
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.
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional ``coin space'' must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of a DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on d-regular, d-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a ``coined'' continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at a spatial search by a DTQW over the two-dimensional (2D) torus (a grid with periodic boundary conditions) of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$, where it was shown that a coined DTQW can search in time O(NlogN), but a standard CTQW takes $\mathrm{\ensuremath{\Omega}}(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2D torus exhibits the O(NlogN) scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in Krovi and Brun, Phys. Rev. A 75, 062332 (2007).
The position density of a "particle" performing a continuous-time quantum walk on the integer lattice, viewed on length scales inversely proportional to the time , converges (as tends to infinity) to a probability distribution that depends on the initial state of the particle. This convergence behavior has recently been demonstrated for the simplest continuous-time random walk [N. Konno, Phys. Rev. E 72, 026113 (2005)]. In this Brief Report, we use a different technique to establish the same convergence for a very large class of continuous-time quantum walks, and we identify the limit distribution in the general case.