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arXiv:1104.0596 · International Journal of Quantum Information

SYMMETRY AND QUANTUM TRANSPORT ON NETWORKS

S. Salimi, Roya Radgohar, M. M. Soltanzadeh

Published Dec 1, 2010• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0219749911006661

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We consider the quantum mechanical transport of (coherent) excitons on small-world networks (SWNs). The SWNs are built from a one-dimensional ring of N nodes by randomly introducing B additional bonds between them. The exciton dynamics is modeled by continuous-time quantum walks, and we evaluate numerically the ensemble-averaged transition probability to reach any node of the network from the initially excited one. For sufficiently large B we find that the quantum mechanical transport through the SWNs is, first, very fast, given that the limiting value of the transition probability is reached very quickly, and second, that the transport does not lead to equipartition, given that on average the exciton is most likely to be found at the initial node.
Continuous time quantum walks (CTQWs) do not necessarily perform better than their classical counterparts, the continuous time random walks (CTRWs). For one special graph, where a recent analysis showed that in a particular direction of propagation the penetration of the graph is faster by CTQWs than by CTRWs, we demonstrate that in another direction of propagation the opposite is true. In this case a CTQW initially localized at one site displays a slow transport. We furthermore show that when the CTQW's initial condition is a totally symmetric superposition of states of equivalent sites, the transport gets to be much more rapid.
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Abstract We study single-particle quantum transport on multilayer generalized scale-free networks using the continuous-time quantum walk model. Our focus is directed at the average return probability and its long-time average value as measures for the transport efficiency. In the continuous-time model these quantities are completely determined by all the eigenvalues and eigenvectors of the connectivity matrix. For all multilayer networks a nontrivial interplay between good spreading and localization effects is observed. The spreading is enhanced by increasing the number of layers L or the power-law exponent γ of the degree distribution. For our choice of the parameters, namely L (1 ≤ L ≤ 50) or γ (1 ≤ γ ≤ 4), the quantum efficiency is increased by at least one order of magnitude. The topological transition between networks without loops, which corresponds to a single scale-free network layer (L = 1), and networks with loops (L = 2) is the most impactful. Another important change occurs when L gets higher than the average diameter d of the layers, namely a new scaling behavior for random walks and lower fluctuations around the long-time average value for quantum walks. The quantum transport is more sensitive to changes of the minimum allowed degree, K min, than to the maximum allowed degree, K max. The same quantum efficiency is found by varying at least one of the parameters: L, γ, K min, or K max, although the network’s topology is different. The quantum efficiency of all multilayer scale-free networks shows a universal behavior for any size of the layers, more precise, is inversely proportional to the number of layers.