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arXiv:0903.1149 · Journal of Physics A Mathematical and Theoretical

Exact analytical results for quantum walks on star graphs

X. P. Xu

Published Feb 20, 2009• 46 citations
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Abstract

In this paper, we study coherent exciton transport of continuous-time quantum walks on star graphs.Exact analytical results of the transition probabilities are obtained by means of the Gram-Schmidt orthonormalization of the eigenstates.Our results show that the coherent exciton transport displays perfect revivals and strong localization on the initial node.When the initial excitation starts at the central node, the transport on a star graph is equivalent to the transport on a complete graph of the same size.

DOI
10.1088/1751-8113/42/11/115205

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In this paper, we study coherent exciton transport of continuous-time quantum walks on star graphs.Exact analytical results of the transition probabilities are obtained by means of the Gram-Schmidt orthonormalization of the eigenstates.Our results show that the coherent exciton transport displays perfect revivals and strong localization on the initial node.When the initial excitation starts at the central node, the transport on a star graph is equivalent to the transport on a complete graph of the same size.
We study the coherent exciton transport on Apollonian networks generated by simple iterative rules. The coherent exciton dynamics is modeled by continuous-time quantum walks and we calculate the transition probabilities between two nodes of the networks. We find that the transport depends on the initial nodes of the excitation. For networks up to the second generation the coherent transport shows perfect recurrences when the initial excitation starts at the central node. For networks of higher generation, the transport only shows partial revivals. Moreover, we find that the excitation is most likely to be found at the initial nodes while the coherent transport to other nodes has a very low probability. In the long time limit, the transition probabilities show characteristic patterns with identical values of limiting probabilities. Finally, the dynamics of quantum transport are compared with the classical transport modeled by continuous-time random walks.
The coherent exciton transport on a class of deterministic and random scale-free networks (DSFNs and RSFNs) generated by simple rules is studied in this paper. The coherent exciton dynamics is modeled by continuous-time quantum walks, and we calculate the transition probabilities between two nodes of the networks. We find that the transport depends on the initial nodes of the excitation. For DSFNs, the probabilities of finding the excitation at the initial central nodes are nearly periodic, in contrast to the flat behavior found for RSFNs. In the long time limit, the transition probabilities on DSFNs show characteristic patterns with identical values. For RSFNs, we find that the excitation is most likely to be found at the initial nodes with high connectivity. All these features of quantum transport are significantly different from those of the classical transport modeled by continuous-time random walks.
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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.
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.
This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter $\ensuremath{\alpha}$. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes NNN, time ttt, and the model parameter $\ensuremath{\alpha}$. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerical simulations of quantum walks on circulant graphs show complex interference phenomena and how complete suppression of transport is achieved near $\ensuremath{\alpha}=\ensuremath{\pi}/2$. By proving two mirror symmetries around $\ensuremath{\alpha}=0$ and $\ensuremath{\pi}/2$ we show that these quantum walks have a period of $\ensuremath{\pi}$ in $\ensuremath{\alpha}$. We show that undirected edges lose their effect on the quantum walk at $\ensuremath{\alpha}=\ensuremath{\pi}/2$ and present non-bipartite graphs that exhibit suppression of transport. Finally, we analytically compute the Hamiltonians of quantum walks on the directed ring graph.