Journal of Statistical Physics

, Volume 171, Issue 2, pp 207–219 | Cite as

A Spectral Analysis of Discrete-Time Quantum Walks Related to the Birth and Death Chains

  • Choon-Lin Ho
  • Yusuke IdeEmail author
  • Norio Konno
  • Etsuo Segawa
  • Kentaro Takumi


In this paper, we consider a spectral analysis of discrete time quantum walks on the path. For isospectral coin cases, we show that the time averaged distribution and stationary distributions of the quantum walks are described by the pair of eigenvalues of the coins as well as the eigenvalues and eigenvectors of the corresponding random walks which are usually referred as the birth and death chains. As an example of the results, we derive the time averaged distribution of so-called Szegedy’s walk which is related to the Ehrenfest model. It is represented by Krawtchouk polynomials which is the eigenvectors of the model and includes the arcsine law.


Quantum walk Birth and death chain Ehrenfest model Krawtchouk polynomials 



We thank the anonymous referees for their careful reading of our manuscript and their fruitful comments and suggestions. C.L.H. was supported in part by the Ministry of Science and Technology (MoST) of the Republic of China under Grants MoST 105-2112-M-032-003. Y.I. was supported by the Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science (Grant No. 16K17652). N.K. was supported by the Grant-in-Aid for Challenging Exploratory Research of Japan Society for the Promotion of Science (Grant No. 15K13443). E.S. was supported by the Grant-in-Aid for Young Scientists (B) and the Grant-in-Aid for Scientific Research (B) of Japan Society for the Promotion of Science (Grant No. 16K17637, 16H03939).


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Authors and Affiliations

  • Choon-Lin Ho
    • 1
  • Yusuke Ide
    • 2
    Email author
  • Norio Konno
    • 3
  • Etsuo Segawa
    • 4
  • Kentaro Takumi
    • 3
  1. 1.Department of PhysicsTamkang UniversityNew Taipei CityTaiwan, ROC
  2. 2.Department of Information Systems Creation, Faculty of EngineeringKanagawa UniversityYokohamaJapan
  3. 3.Department of Applied Mathematics, Faculty of EngineeringYokohama National UniversityYokohamaJapan
  4. 4.Graduate School of Information ScienceTohoku UniversitySendaiJapan

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