SYMMETRY AND QUANTUM TRANSPORT ON NETWORKS

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.

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Localization of quantum walks on a deterministic recursive tree

Quantum Information and Computation ◽

10.26421/qic11.3-4-5 ◽

2011 ◽

Vol 11 (3&4) ◽

pp. 253-261

Author(s):

Xin-Ping Xu

Keyword(s):

Continuous Time ◽

Infinite System ◽

Quantum Walks ◽

Theoretical Interpretation ◽

Initial Node ◽

Recursive Tree ◽

Laplace Matrix ◽

Return Probability ◽

Approximate Form ◽

Time Quantum

Localization of quantum walks can be characterized by the return probability, i.e., the probability for the walker returning to its original site. In this paper, we consider localization of continuous-time quantum walks in terms of return probability on a deterministic recursive tree, which is generated by adding one node and connecting it to each node of the existing tree recursively. We obtain an approximate form for the return probability using the complete eigenvalues and eigenstates of Laplace matrix of the structure. It is found that the return probability depends on the initial node of the excitation. When the walk starts at the central nodes, the return probability converges to a constant value even in the limit of infinite system, in contrast to an exponential decay of the return probability if the walk starts at the outlying nodes. We also observe a bipartite structure for the distribution of return probability, and provide theoretical interpretation for all our findings. Our results suggest that quantum walks display significant localization and well-bedded structure of return probability on heterogeneous trees.

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Transport Efficiency of Continuous-Time Quantum Walks on Graphs

Entropy ◽

10.3390/e23010085 ◽

2021 ◽

Vol 23 (1) ◽

pp. 85

Author(s):

Luca Razzoli ◽

Matteo G. A. Paris ◽

Paolo Bordone

Keyword(s):

Transport Properties ◽

Continuous Time ◽

Quantum Walk ◽

Quantum Walks ◽

Transport Processes ◽

Initial State ◽

Transport Efficiency ◽

Harvesting Systems ◽

Time Quantum ◽

Modeling Transport

Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex that is accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but, in general, they are uncorrelated.

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Single-point position and transition defects in continuous time quantum walks

Scientific Reports ◽

10.1038/srep13585 ◽

2015 ◽

Vol 5 (1) ◽

Cited By ~ 6

Author(s):

Z. J. Li ◽

J. B. Wang

Keyword(s):

Continuous Time ◽

Single Point ◽

Quantum Walks ◽

Point Position ◽

Time Quantum

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Marking Vertices to Find Graph Isomorphism Mapping Based on Continuous-Time Quantum Walk

Entropy ◽

10.3390/e20080586 ◽

2018 ◽

Vol 20 (8) ◽

pp. 586 ◽

Cited By ~ 1

Author(s):

Xin Wang ◽

Yi Zhang ◽

Kai Lu ◽

Xiaoping Wang ◽

Kai Liu

Keyword(s):

Polynomial Time ◽

Continuous Time ◽

Graph Isomorphism ◽

Quantum Walk ◽

Isomorphism Problem ◽

Quantum Walks ◽

Regular Graphs ◽

Structure Preserving ◽

Topological Structures ◽

Time Quantum

The isomorphism problem involves judging whether two graphs are topologically the same and producing structure-preserving isomorphism mapping. It is widely used in various areas. Diverse algorithms have been proposed to solve this problem in polynomial time, with the help of quantum walks. Some of these algorithms, however, fail to find the isomorphism mapping. Moreover, most algorithms have very limited performance on regular graphs which are generally difficult to deal with due to their symmetry. We propose IsoMarking to discover an isomorphism mapping effectively, based on the quantum walk which is sensitive to topological structures. Firstly, IsoMarking marks vertices so that it can reduce the harmful influence of symmetry. Secondly, IsoMarking can ascertain whether the current candidate bijection is consistent with existing bijections and eventually obtains qualified mapping. Thirdly, our experiments on 1585 pairs of graphs demonstrate that our algorithm performs significantly better on both ordinary graphs and regular graphs.

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