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Discrete-time quantum walk on the Cayley graph of the dihedral group

Authors
  • Dai, Wenjing1
  • Yuan, Jiabin1
  • Li, Dan1
  • 1 Nanjing University of Aeronautics and Astronautics, College of Computer Science and Technology, Nanjing, 211106, China , Nanjing (China)
Type
Published Article
Journal
Quantum Information Processing
Publisher
Springer-Verlag
Publication Date
Oct 24, 2018
Volume
17
Issue
12
Identifiers
DOI: 10.1007/s11128-018-2101-9
Source
Springer Nature
Keywords
License
Yellow

Abstract

The finite dihedral group generated by one rotation and one flip is the simplest case of the non-Abelian group. Cayley graphs are diagrammatic counterparts of groups. In this paper, much attention is given to the Cayley graph of the dihedral group. Considering the characteristics of the elements in the dihedral group, we conduct the model of discrete-time quantum walk on the Cayley graph of the dihedral group by special coding mode. This construction makes Fourier transformation be used to carry out spectral analysis of the dihedral quantum walk, i.e. the non-Abelian case. Furthermore, the relation between quantum walk without memory on the Cayley graph of the dihedral group and quantum walk with memory on a cycle is discussed, so that we can explore the potential of quantum walks without and with memory. Here, the numerical simulation is carried out to verify the theoretical analysis results and other properties of the proposed model are further studied.

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