Quantum Information Processing

, Volume 14, Issue 9, pp 3179–3191 | Cite as

Moments of coinless quantum walks on lattices

  • Raqueline Azevedo Medeiros SantosEmail author
  • Renato Portugal
  • Stefan Boettcher


The properties of the coinless quantum-walk model have not been as thoroughly analyzed as those of the coined model. Both evolve in discrete time steps, but the former uses a smaller Hilbert space, which is spanned merely by the site basis. Besides, the evolution operator can be obtained using a process of lattice tessellation, which is very appealing. The moments of the probability distribution play an important role in the context of quantum walks. The ballistic behavior of the mean square displacement indicates that quantum-walk-based algorithms are faster than random-walk-based ones. In this paper, we obtain analytical expressions for the moments of the coinless model on d-dimensional lattices by employing the methods of Fourier transforms and generating functions. The mean square displacement for large times is explicitly calculated for the one- and two-dimensional lattices, and using optimization methods, the parameter values that give the largest spread are calculated and compared with the equivalent ones of the coined model. Although we have employed asymptotic methods, our approximations are accurate even for small numbers of time steps.


Coinless quantum walks Moments Mean square displacement Standard deviation 



RAMS acknowledges financial support from Capes-Faperj E-45/2013. RP thanks Faperj (Grant No. E-26/102.350/2013) and CNPq (Grant Nos. 304709/2011-5, 474143/2013-9, and 400216/2014-0). SB acknowledges financial support from the US National Science Foundation through Grant DMR-1207431.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Raqueline Azevedo Medeiros Santos
    • 1
    Email author
  • Renato Portugal
    • 1
    • 2
  • Stefan Boettcher
    • 3
  1. 1.Laboratório Nacional de Computação CientíficaPetrópolisBrazil
  2. 2.Universidade Católica de PetrópolisPetrópolisBrazil
  3. 3.Department of PhysicsEmory UniversityAtlantaUSA

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