Quantum Walks on Infinite Graphs

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


Quantum walks on the line were introduced in Sect. 3.2 in order to highlight some features, which are strikingly different from the classical random walks. In this Chapter, we present in detail the analytical calculation of the state of quantum walks on the line. This calculation is a model for the study of quantum walks on many types of graphs. The Fourier transform is the key to the success of this calculation.


Quantum Walk Infinite Graphs Classical Random Walk Hadamard Coin Grover Coin 
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  1. 7.
    Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of 33th STOC, pp. 60–69. ACM, New York (2001)Google Scholar
  2. 19.
    Carteret, H.A., Ismail, M.E.H., Richmond, B.: Three routes to the exact asymptotics for the one-dimensional quantum walk. J. Phys A: Math. General 36(33), 8775–8795 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 44.
    Konno, N.: Quantum random walks in one dimension. Quant. Inform. Process. 1(5), 345–354 (2002)MathSciNetCrossRefGoogle Scholar
  4. 45.
    Košík, J.: Two models of quantum random walk. Cent. Eur. J. Phys. 4, 556–573 (2003)Google Scholar
  5. 49.
    Mackay, T.D., Bartlett, S.D., Stephenson, L.T., Sanders, B.C.: Quantum walks in higher dimensions. J. Phys. A: Math. General 35(12), 2745 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 56.
    Marquezino, F.L., Portugal, R.: The QWalk simulator of quantum walks. Comput. Phys. Commun. 179(5), 359–369 (2008), arXiv:0803.3459Google Scholar
  7. 63.
    Nayak, A., Vishwanath, A.: Quantum walk on a line. DIMACS Technical Report 2000-43, quant-ph/0010117 (2000)Google Scholar
  8. 77.
    Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5(1), 83 (2003), quant-ph/0304204Google Scholar
  9. 79.
    Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Information Processing, pp. 1–92 (2012), arXiv:1201.4780Google Scholar
  10. 80.
    Venegas-Andraca, S.E.: Quantum Walks for Computer Scientists. Morgan and Claypool Publishers, San Rafael (2008)Google Scholar
  11. 82.
    Štefaňák, M.: Interference phenomena in quantum information. PhD thesis, Czech Technical University (2010), arXiv:1009.0200Google 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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