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Journal of Physics A Mathematical and Theoretical

Quantum transport on modified multilayered spiderwebs

Cássio Macêdo Maciel, Walter T. Strunz, Mircea Daniel Galiceanu

Published Oct 16, 2018• 5 citations
Publisher

Abstract

Abstract We consider continuous-time quantum walks (CTQWs) on modified multilayered spiderwebs (MMSs) and their application to quantum transport. Quantum transport efficiency is determined in terms of the exact and the average return probabilities and it is systematically studied for three kind of MMSs. First, we consider a single layer of modified spiderwebs and we introduce the parameter p , which controls the transition from a pure Cayley tree ( p = 0.0) to a complete spiderweb ( p = 1.0). Topologically, this transition corresponds to adding with probability p more links between nearest neighboring nodes from the same generation, when we start from a Cayley tree. By doing so, we observe an increase in the quantum efficiency and we notice that for all generation numbers its highest value is encountered when only one link is missing from each generation. In the second case we stack more modified spiderwebs on top of each other and interconnect them, obtaining an MMS network. In this case the quantum efficiency is further increased by more than two orders of magnitude. In the third case, we remove some interconnecting links between nearest neighboring layers with probability 1 − q . For these kind of networks we encounter that the highest quantum efficiency does not correspond to q = 1.0, but to , more precise when we remove a link between successive layers. For these networks and for a proper choice of parameters the quantum efficiency is further increased by several orders of magnitude.

DOI
10.1088/1751-8121/aae8ab

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Abstract We consider continuous-time quantum walks (CTQWs) on modified multilayered spiderwebs (MMSs) and their application to quantum transport. Quantum transport efficiency is determined in terms of the exact and the average return probabilities and it is systematically studied for three kind of MMSs. First, we consider a single layer of modified spiderwebs and we introduce the parameter p , which controls the transition from a pure Cayley tree ( p = 0.0) to a complete spiderweb ( p = 1.0). Topologically, this transition corresponds to adding with probability p more links between nearest neighboring nodes from the same generation, when we start from a Cayley tree. By doing so, we observe an increase in the quantum efficiency and we notice that for all generation numbers its highest value is encountered when only one link is missing from each generation. In the second case we stack more modified spiderwebs on top of each other and interconnect them, obtaining an MMS network. In this case the quantum efficiency is further increased by more than two orders of magnitude. In the third case, we remove some interconnecting links between nearest neighboring layers with probability 1 − q . For these kind of networks we encounter that the highest quantum efficiency does not correspond to q = 1.0, but to , more precise when we remove a link between successive layers. For these networks and for a proper choice of parameters the quantum efficiency is further increased by several orders of magnitude.
We consider continuous-time quantum walks (CTQWs) on multilayer dendrimer networks (MDs) and their application to quantum transport. A detailed study of properties of CTQWs is presented and transport efficiency is determined in terms of the exact and average return probabilities. The latter depends only on the eigenvalues of the connectivity matrix, which even for very large structures allows a complete analytical solution for this particular choice of network. In the case of MDs we observe an interplay between strong localization effects, due to the dendrimer topology, and good efficiency from the linear segments. We show that quantum transport is enhanced by interconnecting more layers of dendrimers.
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.
We study the transport properties on honeycomb networks motivated by graphene structures by using the continuous-time quantum walk (CTQW) model. For various relevant topologies we consider the average return probability and its long-time average as measures for the transport efficiency. These quantities are fully determined by the eigenvalues and the eigenvectors of the connectivity matrix of the network. For all networks derived from graphene structures we notice a nontrivial interplay between good spreading and localization effects. Flat graphene with similar number of hexagons along both directions shows a decrease in transport efficiency compared to more one-dimensional structures. This loss can be overcome by increasing the number of layers, thus creating a graphite network, but it gets less efficient when rolling up the sheets so that a nanotube structure is considered. We found peculiar results for honeycomb networks constructed from square graphene, i.e. the same number of hexagons along both directions of the graphene sheet. For these kind of networks we encounter significant differences between networks with an even or odd number of hexagons along one of the axes.
We consider quantum transport on generalized scale-free networks (GSFNs) in the continuous-time quantum walk (CTQW) model. The efficiency of the transport is monitored through the exact and the average return probabilities. In this model these probabilities are fully determined by the eigenvalues and eigenvectors of the connectivity matrix. In the case of GSFNs we observe a nontrivial interplay between strong localization effects, due to starlike segments, and good spreading because of the linear segments. We show that the quantum transport on GSFNs can be increased by varying the minimum or the maximum allowed degrees, i.e., the limiting number of links emerging from every node. The same quantum efficiency is reached by considering various combinations of the construction parameters of the network, which normally show different topological features.
In this paper, we consider continuous-time quantum walks (CTQWs) on a one-dimensional ring lattice of N nodes in which every node is connected to its 2m nearest neighbors ( m on either side). In the framework of the Bloch function ansatz, we calculate the space-time transition probabilities between two nodes of the lattice. We find that the transport of CTQWs between two different nodes is faster than that of the classical continuous-time random walks (CTRWs). The transport speed, which is defined by the ratio of the shortest path length and propagating time, increases with the connectivity parameter m for both CTQWs and CTRWs. For fixed parameter m , the transport of CTRWs gets slower with the increase of the shortest distance while the transport (speed) of CTQWs turns out to be a constant value. In the long-time limit, depending on the network size N and connectivity parameter m , the limiting probability distributions of CTQWs show various patterns. When the network size N is an even number, the probability of being at the original node differs from that of being at the opposite node, which also depends on the precise value of parameter m .
We study the coherent exciton transport of continuous-time quantum walks (CTQWs) on Erdos-Renyi networks. The Erdos-Renyi network of N nodes is constructed by connecting every pair of nodes with probability ppp. We numerically calculate the ensemble averaged transition probability of quantum transport between two nodes of the networks. For finite networks, we find that the limiting transition probability is reached very quickly. For infinite networks whose spectral density follows the semicircle law, the efficiencies of the classical and quantum-mechanical transport are compared on networks of different average degree. In the long time limiting, we consider the distribution of the ensemble averaged transition probabilities, and show that there is a high probability to find the exciton at the initial node. Such high return probability almost do not alter in a wide range of connection probability p but increases rapidly when the network approaches to be fully connected. For networks whose topology is not extremely connected, the return probability is inversely proportional to the network size N. Furthermore, the transport dynamics are compared with that on a random graph model in which the degree of each node equals to the average degree of the Erdos-Renyi networks.