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Discrete-Time Quantum Walk with Memory on the Cayley Graph of the Dihedral Group

Abstract

By adding one-step memory to enrich the model of discrete-time quantum walk on the Caylay graph of the dihedral group, the model of quantum walk with memory on the Cayley graph of the dihedral group is constructed. The Fourier Transform is used to analyze the walk, and a formula for probability distribution and time-average probability distribution is given. According to the one-to-one correspondence between quantum walk with memory on a regular graph and quantum walk without memory on the corresponding line digraph, the diagrammatic counterpart of quantum walk with one-step memory on the Cayley graph of the dihedral group is presented. Moreover, basic probabilistic properties of the proposed model are further studied using numerical simulation method. Furthermore, the similarities and differences are discussed in comparison to the memoryless model.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China(Grant Nos. 61571226, 61701229, 61572053, 61702367), Natural Science Foundation of Jiangsu Province, China (Grant No. BK20170802), Postdoctoral Science Foundation funded Project of China (Grant Nos. 2018M630557, 2018T110499), Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 1701139B), the Beijing Natural Science Foundation (Grant No. 4162005), the Research Project of Tianjin Municipal Commission of Education(Grant No. 2017KJ033).

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Correspondence to Wenjing Dai.

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Dai, W., Yuan, J. & Li, D. Discrete-Time Quantum Walk with Memory on the Cayley Graph of the Dihedral Group. Int J Theor Phys 59, 10–28 (2020) doi:10.1007/s10773-019-04257-7

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Keywords

  • Discrete-time quantum walk
  • The Caylay graph of the dihedral group
  • With memory
  • Fourier transform
  • Probability distribution
  • Regular graph
  • Line digraph
  • Numerical simulation