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arXiv:0810.1184 · Journal of Physics A Mathematical and Theoretical

Dynamics of continuous-time quantum walks in restricted geometries

E Agliari, A. Blumen, O Mülken

Published Oct 7, 2008• 45 citations
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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.

DOI
10.1088/1751-8113/41/44/445301

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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.
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 model quantum transport, described by continuous-time quantum walks (CTQWs), on deterministic Sierpinski fractals, differentiating between Sierpinski gaskets and Sierpinski carpets, along with their dual structures. The transport efficiencies are defined in terms of the exact and the average return probabilities, as well as by the mean survival probability when absorbing traps are present. In the case of gaskets, localization can be identified already for small networks (generations). For carpets, our numerical results indicate a trend towards localization, but only for relatively large structures. The comparison of gaskets and carpets further implies that, distinct from the corresponding classical continuous-time random walk, the spectral dimension does not fully determine the evolution of the CTQW.
A Multilayer network is a potent platform that paves the way for the study of the interactions among entities in various networks with multiple types of relationships. This study explores the dynamics of discrete-time quantum walks on a multilayer network. We derive a recurrence formula for the coefficients of the wave function of a quantum walker on an undirected graph with a finite number of nodes. By extending this formula to include extra layers, we develop a simulation model to describe the time evolution of the quantum walker on a multilayer network. The time-averaged probability and the return probability of the quantum walker are studied with Fourier, and Grover walks on multilayer networks. Furthermore, we analyze the impact of decoherence on quantum transport, shedding light on how environmental interactions may impact the behavior of quantum walkers on multilayer network structures.
One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a lattice or graph containing a marked site. In most conceivable physical applications, however, one would rather expect to have multiple interacting particles. In bosonic systems at zero temperature, the dynamics would be well-described by a discrete non-linear Schrodinger equation. In this thesis we investigate the role of non-linearity in determining the efficiency of the spatial search algorithm within the quantum walk model, for the complete graph. Our analytical results indicate that the search time for this non-linear quantum search scales with size of the database N like square root of N, equivalent to linear spatial search time. The analytical results will be compared with numerical calculations of multiple interacting quantum walkers.
We study the classical and quantum transport processes on some finite networks and model them by continuous-time random walks (CTRW) and continuous-time quantum walks (CTQW), respectively. We calculate the classical and quantum transition probabilities between two nodes of the network. We numerically show that there is a high probability to find the walker at the initial node for CTQWs on the underlying networks due to the interference phenomenon, even for long times. To get global information (independent of the starting node) about the transport efficiency, we average the return probability over all nodes of the network. We apply the decay rate and the asymptotic value of the average of the return probability to evaluate the transport efficiency. Our numerical results prove that the existence of the symmetry in the underlying networks makes quantum transport less efficient than the classical one. In addition, we find that the increasing of the symmetry of these networks decreases the efficiency of quantum transport on them.
We study the transport properties of continuous-time quantum walks (CTQWs) over finite two-dimensional structures with a given number of randomly placed bonds and with different aspect ratios (ARs). Here, we focus on the transport from, say, the left side to the right side of the structure where absorbing sites are placed. We do so by analyzing the long-time average of the survival probability of CTQWs. We compare the results to the classical continuous-time random walk case (CTRW). For small ARs (landscape configurations) we observe only small differences between the quantum and the classical transport properties, i.e., roughly the same number of bonds is needed to facilitate the transport. However, with increasing ARs (portrait configurations) a much larger number of bonds is needed in the CTQW case than in the CTRW case. While for CTRWs the number of bonds needed decreases when going from small ARs to large ARs, for CTQWs this number is large for small ARs, has a minimum for the square configuration, and increases again for increasing ARs. We explain our findings by analyzing the average eigenstates of the corresponding structures: The participation ratios allow us to distinguish between localized and nonlocalized (average) eigenstates. In particular, for large ARs we find for CTQWs that the eigenstates are localized for bond numbers exceeding the bond numbers needed to facilitate transport in the CTRW case. Thus, a rather large number of bonds is needed in order for quantum transport to be efficient for large ARs.