Quantum walk with quadratic position-dependent phase defects

  • Umer Farooq
  • Abdullah S. Alshammari
  • Najeh RekikEmail author


A comprehensive study of the property of one-dimensional quantum walks, via position distribution of the walker, is adopted herein by considering position-dependent quadratic phase defect. We have explored the origins of this property by introducing a designated position quadratic phase defect conditional shift operator for a discrete-time quantum walk. Numerical simulations conclude that the revival of the walker depends on the phase modulation parameter and the number of steps. In addition, we have found that the localization effect appears after the threshold point where the amplitude of the walker reaches its maximum value, at a constant interval corresponding to the revival period. Furthermore, numerical results show that for rational parabolic coefficients, \(2\pi \frac{q}{p}\), of the phase defect profile, revivals occur with period 2p for odd p and period p for even p. Furthermore, the period of revival as a function of p is the same as in the case of a linear (instead of quadratic) phase gradient, which was previously investigated in many studies [for instance, see Wójcik et al. (Phys Rev Lett 93:180601, 2004) and Cedzich et al. (Phys Rev Lett 111:160601, 2013)]. The results of this approach therefore shed light on the analysis of discrete quantum processes and the potential relevant for physical implementations of quantum computing with various mesoscopic systems.


Quantum information One-dimensional quantum walks Quadratic phase defect Revival 



Dr. Najeh Rekik thanks the Deanship of Scientific Research, University of Ha’il, Kingdom of Saudi Arabia, for the financial support under Grant No. SP14005. The authors would also like to thank Dr. Ryan Zaari (Department of Chemistry, University of Alberta) for the careful reading of the manuscript and the fruitful comments and suggestions.


  1. 1.
    Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1 (1943)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687 (1993)ADSCrossRefGoogle Scholar
  3. 3.
    Guillotin-Plantard, N., Schott, R.: Dynamic Random Walks: Theory and Application. Elsevier, Amsterdam (2006)zbMATHCrossRefGoogle Scholar
  4. 4.
    Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507–518 (2003)zbMATHCrossRefGoogle Scholar
  5. 5.
    Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)ADSCrossRefGoogle Scholar
  6. 6.
    Meyer, D.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85, 551–574 (1996)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Aharonov, D., Ambainis. A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of STOC’01, pp. 50–59Google Scholar
  8. 8.
    Bachelier, L.: Theory of speculation. Ann. Sci. Ecole Norm. Super. 17, 21 (1900)zbMATHCrossRefGoogle Scholar
  9. 9.
    Childs, A.M.: In: Proceedings of ACM Symposium on Theory of Computing (STOC 2003) pp. 59–68 (2003)Google Scholar
  10. 10.
    Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307–327 (2003)ADSCrossRefGoogle Scholar
  11. 11.
    Berry, S.D., Wang, J.B.: Quantum-walk-based search and centrality. Phys. Rev. A 82, 042333 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Franco, C., McGettrick, D., Busch, M.: Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin. Phys. Rev. Lett. 106, 080502 (2011)CrossRefGoogle Scholar
  13. 13.
    Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Childs, A.M., Gosset, D., Webb, Z.: Universal computation by multiparticle quantum walk. Science 339, 791–794 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lovett, N.B., Cooper, S., Everitt, M., Trevers, M., Kendon, V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81, 042330 (2010)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Lloyd, S.: Quantum coherence in biological systems. J. Phys. Conf. Ser. 302, 012037 (2011)CrossRefGoogle Scholar
  17. 17.
    Hoyer, S., Sarovar, M., Whaley, K.B.: Limits of quantum speedup in photosynthetic light harvesting. New J. Phys. 12, 065041 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958)ADSCrossRefGoogle Scholar
  19. 19.
    Mohseni, P.M., Rebentrost, P., Lloyd, S., Aspuru-Guzik, A.: Environment-assisted quantum walks in energy transfer of photosynthetic complexes. J. Chem. Phys. 129, 174106 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    Ren, J., Chen, T., Zhang, X.: Long-lived quantum speedup based on plasmonic hot spot systems. New J. Phys. 21, 053034 (2019)ADSCrossRefGoogle Scholar
  21. 21.
    Wójcik, A.: Trapping a particle of a quantum walk on the line. Phys. Rev. A 85, 012329 (2012)ADSCrossRefGoogle Scholar
  22. 22.
    Konno, N.: Localization of an inhomogeneous discrete-time quantum walk on the line. Quantum Inf. Process. 9, 405–418 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Shikano, Y., Katsura, H.: Localization and fractality in inhomogeneous quantum walks with self-duality. Phys. Rev. E 82, 031122 (2010)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhang, R., Xue, P., Twamley, J.: One-dimensional quantum walks with single-point phase defects. Phys. Rev. A 89, 042317 (2014)ADSCrossRefGoogle Scholar
  25. 25.
    Schreiber, A.: Decoherence and disorder in quantum walks: from ballistic spread to localization. Phys. Rev. Lett. 106, 180403 (2011)ADSCrossRefGoogle Scholar
  26. 26.
    Crespi, A.: Anderson localization of entangled photons in an integrated quantum walk. Nat. Photonics 7, 322–328 (2013)ADSCrossRefGoogle Scholar
  27. 27.
    Kitagawa, T., Broome, M.A., Fedrizzi, A., Rudner, M.S., Berg, E., Kassal, I., Aspuru-Guzik, A., Demler, E., White, A.G.: Observation of topologically protected bound states in photonic quantum walks. Nat. Commun. 3, 882 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    Wójcik, A., Łuczak, T., Kurzyński, P., Grudka, A., Bednarska, M.: Quasiperiodic dynamics of a quantum walk on the line. Phys. Rev. Lett. 93, 180601 (2004)ADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Sasaki, T., Yamamoto, Y., Koashi, M.: Practical quantum key distribution protocol without monitoring signal disturbance. Nature 509(7501), 475–478 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    Zahringer, F., Kirchmair, G., Gerritsma, R., Solano, E., Blatt, R., Roos, C.F.: Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104, 100503 (2010)ADSCrossRefGoogle Scholar
  31. 31.
    Xue, P., Qin, H., Tang, B.: Trapping photons on the line: controllable dynamics of a quantum walk. Sci. Rep. 4, 4825 (2014)ADSCrossRefGoogle Scholar
  32. 32.
    Schmitz, H., Matjeschk, R., Schneider, Ch., Glueckert, J., Enderlein, M., Huber, T., Schaetz, T.: Quantum walk of a trapped ion in phase space. Phys. Rev. Lett. 103, 090504 (2009)ADSCrossRefGoogle Scholar
  33. 33.
    Cote, R., Russell, A., Eyler, E.E., Gould, P.L.: Quantum random walk with Rydberg atoms in an optical lattice. New J. Phys. 8, 156 (2006)ADSCrossRefGoogle Scholar
  34. 34.
    Karski, M., Forster, L., Choi, J., Steffen, A., Alt, W., Meschede, D., Widera, A.: Quantum walk in position space with single optically trapped atoms. Science 325, 174 (2009)ADSCrossRefGoogle Scholar
  35. 35.
    Do, B., Stohler, M.L., Balasubramanian, S., Elliott, D.S., Eash, C., Fischbach, E., Fischbach, M.A., Mills, A., Zwickl, B.: Experimental realization of a quantum quincunx by use of linear optical elements. J. Opt. Soc. Am. B 22, 499 (2005)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Broome, M.A., Fedrizzi, A., Lanyon, B.P., Kassal, I., Aspuru-Guzik, A., White, A.G.: Discrete single-photon quantum walks with tunable decoherence. Phys. Rev. Lett. 104, 153602 (2010)ADSCrossRefGoogle Scholar
  37. 37.
    Crespi, A., Osellame, R., Ramponi, R., Giovannetti, V., Fazio, R., Sansoni, L., Nicola, F.D., Sciarrino, F., Mataloni, P.: Anderson localization of entangled photons in an integrated quantum walk. Nat. Photonics 7, 322 (2013)ADSCrossRefGoogle Scholar
  38. 38.
    Owens, J.O., Broome, M.A., Biggerstaff, D.N., Goggin, M.E., Fedrizzi, A., Linjordet, T., Ams, M., Marshall, G.D., Twamley, J., Withford, M.J., White, A.G.: Two-photon quantum walks in an elliptical direct-write waveguide array. New J. Phys. 13, 075003 (2011)ADSCrossRefGoogle Scholar
  39. 39.
    Sansoni, L., Sciarrino, F., Vallone, G., Mataloni, P., Crespi, A., Ramponi, R., Osellame, R.: Two-particle bosonic–fermionic quantum walk via integrated photonics. Phys. Rev. Lett. 108, 010502 (2012)ADSCrossRefGoogle Scholar
  40. 40.
    Tang, H., et al.: Experimental two-dimensional quantum walk on a photonic chip. Sci. Adv. 4(5), eaat3174 (2018) ADSCrossRefGoogle Scholar
  41. 41.
    Schreiber, A., Cassemiro, K.N., Potoček, V., Gábris, A., Mosley, P.J., Andersson, E., Jex, I., Silberhorn, C.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. vol. 104, p. 050502 (2010)Google Scholar
  42. 42.
    Peruzzo, A., Lobino, M., Matthews, J.C.F., Matsuda, N., Politi, A., Poulios, K., Zhou, X., Lahini, Y., Ismail, N., Wőrhoff, K., Bromberg, Y., Silberberg, Y., Thompson, M.G., OBrien, J.L.: Quantum walks of correlated photons. Science 329, 1500 (2010)ADSCrossRefGoogle Scholar
  43. 43.
    Schreiber, A., Gábris, A., Rohde, P.P., Laiho, K., Štefaňák, M., Potoček, V., Hamilton, C., Jex, I., Silberhorn, Ch.: A 2D quantum walk simulation of two-particle dynamics. Science 336, 55–58 (2012)ADSCrossRefGoogle Scholar
  44. 44.
    Travaglione, B.C., Milburn, G.J.: Implementing the quantum random walk. Phys. Rev. A 65, 032310 (2002)ADSCrossRefGoogle Scholar
  45. 45.
    Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5, 83 (2003)ADSCrossRefGoogle Scholar
  46. 46.
    Cedzich, C., Rybar, T., Werner, A.H., Alberti, A., Genske, M., Werner, R.F.: Propagation of quantum walks in electric fields. Phys. Rev. Lett. 111, 160601 (2003)CrossRefGoogle Scholar
  47. 47.
    Ramasesh, V.V., Flurin, E., Rudner, M.S., Siddiqi, I., Yao, N.Y.: Direct Probe of Topological Invariants Using Bloch Oscillating Quantum Walks, arXiv:1609.09504 (2016)

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Physics Department, College of ScienceUniversity of Ha’ilHa’ilKingdom of Saudi Arabia
  2. 2.Department of ChemistryUniversity of AlbertaEdmontonCanada

Personalised recommendations