Limiting Distribution and Mixing Time

  • Renato Portugal
Part of the Quantum Science and Technology book series (QST)


In this chapter, we use the notion of quantum walks on finite regular graphs with the goal of analyzing the limiting probability distribution and the mixing time. In finite quantum systems, there is a quasi-periodic pattern in the time evolution, preventing the convergence to a limiting distribution. The quasi-periodic behavior of the quantum state can be obtained from the expression of the eigenvalues of the evolution operator. A possible way to obtain limiting configurations is to define a new distribution called average probability distribution, which evolves stochastically and does not have the quasi-periodic behavior.


Limiting Probability Distribution Mixing Time Quantum Walk Finite Regular Graph Quasi-periodic Pattern 
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  1. 3.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of 33th STOC, pp. 50–59. ACM, New York (2001)Google Scholar
  2. 13.
    Bednarska, M., Grudka, A., Kurzynski, P., Luczak, T., Wójcik, A.: Quantum walks on cycles. Phys. Lett. A 317(1–2), 21–25 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 14.
    Bednarska, M., Grudka, A., Kurzynski, P., Luczak, T., Wójcik, A.: Examples of non-uniform limiting distributions for the quantum walk on even cycles. Int. J. Quant. Inform. 2(4), 453–459 (2004)zbMATHCrossRefGoogle Scholar
  4. 54.
    Marquezino, F.L., Portugal, R., Abal, G.: Mixing times in quantum walks on two-dimensional grids. Phys. Rev. A 82(4), 042341 (2010)MathSciNetADSCrossRefGoogle Scholar
  5. 55.
    Marquezino, F.L., Portugal, R., Abal, G., Donangelo, R.: Mixing times in quantum walks on the hypercube. Phys. Rev. A 77, 042312 (2008)MathSciNetADSCrossRefGoogle Scholar
  6. 59.
    Moore, C., Russell, A.: Quantum walks on the hypercube. In: Rolim, J.D.P., Vadhan, S. (eds.) Proceedings of Random 2002, pp. 164–178. Springer, Cambridge (2002)Google Scholar
  7. 60.
    Moore, C., Mertens, S.: The Nature of Computation. Oxford University Press, New York (2011)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Renato Portugal
    • 1
  1. 1.Department of Computer ScienceNational Laboratory of Scientific Computing (LNCC)PetrópolisBrazil

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