Quantum Information Processing

, Volume 11, Issue 5, pp 1287–1299 | Cite as

Quantum walk on distinguishable non-interacting many-particles and indistinguishable two-particle

  • C. M. ChandrashekarEmail author
  • Th. Busch


We present an investigation of many-particle quantum walks in systems of non-interacting distinguishable particles. Along with a redistribution of the many-particle density profile we show that the collective evolution of the many-particle system resembles the single-particle quantum walk evolution when the number of steps is greater than the number of particles in the system. For non-uniform initial states we show that the quantum walks can be effectively used to separate the basis states of the particle in position space and grouping like state together. We also discuss a two-particle quantum walk on a two-dimensional lattice and demonstrate an evolution leading to the localization of both particles at the center of the lattice. Finally we discuss the outcome of a quantum walk of two indistinguishable particles interacting at some point during the evolution.


Distinguishable many particles Indistinguishable two-particles Quantum walks Discrete-time quantum walk 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Riazanov, G.V.: The Feynman path integral for the Dirae equation, Zh. Eksp. Teor. Fiz. 33, 1437 (1958), [Soviet Phys. JETP 6, 1107–1113 (1958)]Google Scholar
  2. 2.
    Feynman R.P., Hibbs A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)zbMATHGoogle Scholar
  3. 3.
    Aharonov Y., Davidovich L., Zagury N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)ADSCrossRefGoogle Scholar
  4. 4.
    Meyer D.A.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85, 551–574 (1996)ADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceeding of the 33rd ACM Symposium on Theory of Computing, p. 60. ACM Press, New York (2001)Google Scholar
  7. 7.
    Nayak, A., Vishwanath, A.: Quantum Walk on the Line, DIMACS Technical Report, No. 2000-43 (2001); arXiv:quant-ph/0010117Google Scholar
  8. 8.
    Ambainis A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1(4), 507–518 (2003)zbMATHCrossRefGoogle Scholar
  9. 9.
    Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by a quantum walk. In: Proceedings of the 35th ACM Symposium on Theory of Computing, p. 59. ACM Press, New York (2003)Google Scholar
  10. 10.
    Shenvi N., Kempe J., Birgitta Whaley K.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)ADSCrossRefGoogle Scholar
  11. 11.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 1099-1108. AMC Press, New York (2005)Google Scholar
  12. 12.
    Chandrashekar C.M., Laflamme R.: Quantum phase transition using quantum walks in an optical lattice. Phys. Rev. A 78, 022314 (2008)ADSCrossRefGoogle Scholar
  13. 13.
    Oka T., Konno N., Arita R., Aoki H.: Breakdown of an electric-field driven system: a mapping to a quantum walk. Phys. Rev. Lett. 94, 100602 (2005)ADSCrossRefGoogle Scholar
  14. 14.
    Engel G.S. et al.: Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature 446, 782–786 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    Mohseni M., Rebentrost P., Lloyd S., Aspuru-Guzik A.: Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys. 129, 174106 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    Chandrashekar, C.M., Goyal, S.K., Banerjee, S.: Entanglement generation in spatially separated systems using quantum walk arXiv:1005.3785 (2010)Google Scholar
  17. 17.
    Kitagawa T., Rudner M.S., Berg E., Demler E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Du J., Li H., Xu X., Shi M., Wu J., Zhou X., Han R.: Experimental implementation of the quantum random-walk algorithm. Phys. Rev. A 67, 042316 (2003)ADSCrossRefGoogle Scholar
  19. 19.
    Ryan C.A., Laforest M., Boileau J.C., Laflamme R.: Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor. Phys. Rev. A 72, 062317 (2005)ADSCrossRefGoogle Scholar
  20. 20.
    Perets H.B., Lahini Y., Pozzi F., Sorel M., Morandotti R., Silberberg Y.: Realization of quantum walks with negligible decoherence in waveguide lattices. Phys. Rev. Lett. 100, 170506 (2008)ADSCrossRefGoogle Scholar
  21. 21.
    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
  22. 22.
    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
  23. 23.
    Karski K., Foster L., Choi J.-M., Steffen A., Alt W., Meschede D., Widera A.: Quantum walk in position space with single optically trapped atoms. Science 325, 174–177 (2009)ADSCrossRefGoogle Scholar
  24. 24.
    Schreiber A., Cassemiro K.N., Potocek V., Gabris A., Mosley P., Andersson E., Jex I., Silberhorn Ch.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104, 05502 (2010)CrossRefGoogle Scholar
  25. 25.
    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
  26. 26.
    Peruzzo A., Lobino M., Matthews J.C.F., Matsuda N., Politi A., Poulios K., Zhou X.-Q., Lahini Y., Ismail N., Wrhoff K., Bromberg Y., Silberberg Y., Thompson M.G., OBrien J.L.: Quantum walks of correlated photons. Science 329, 1500–1503 (2010)ADSCrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    Konno N.: Quantum random walks in one dimension. Quantum Inf. Process. 1(5), 345–354 (2002)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Chandrashekar C.M., Srikanth R., Laflamme R.: Optimizing the discrete time quantum walk using a SU(2) coin. Phys. Rev. A 77, 032326 (2008)ADSCrossRefGoogle Scholar
  30. 30.
    Mayer K., Tichy M.C., Mintert F., Konrad T., Buchleitner A.: Counting statistics of many-particle quantum walks. Phys. Rev. A 83, 062307 (2011)ADSCrossRefGoogle Scholar
  31. 31.
    Rohde P.P., Schreiber A., Stefanak M., Jex I., Silberhorn C.: Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation. New J. Phys. 13, 013001 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    Goyal S.K., Chandrashekar C.M.: Spatial entanglement using a quantum walk on a many-body system. J. Phys. A: Math. Theor. 43, 235303 (2010)MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Mandel O., Greiner M., Widera A., Rom T., Hänsch T.W., Bloch I.: Coherent transport of neutral atoms in spin-dependent optical lattice potentials. Phys. Rev. Lett. 91, 010407 (2003)ADSCrossRefGoogle Scholar
  34. 34.
    Duan L.-M., Demler E., Lukin M.D.: Controlling spin exchange interactions of ultracold atoms in optical lattices. Phys. Rev. Lett. 91, 090402 (2003)ADSCrossRefGoogle Scholar
  35. 35.
    Jaksch D.: Optical lattices, ultracold atoms and quantum information processing. Contemp. Phys. 45(5), 367–381 (2004)ADSCrossRefGoogle Scholar
  36. 36.
    Stefanak M., Kiss T., Jex I., Mohring B.: The meeting problem in the quantum walk. J. Phys. A: Math. Gen. 39, 14965–14983 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 37.
    Knight P.L., Roldan E., Sipe J.E.: Quantum walk on the line as an interference phenomenon. Phys. Rev. A 68, 020301(R) (2003)ADSCrossRefGoogle Scholar
  38. 38.
    Omar Y., Paunkovic N., Sheridan L., Bose S.: Quantum walk on a line with two entangled particles. Phys. Rev. A 74, 042304 (2006)MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    Gamble J.K., Friesen M., Zhou D., Joynt R., Coppersmith S.N.: Two-particle quantum walks applied to the graph isomorphism problem. Phys. Rev. A 81, 052313 (2010)ADSCrossRefGoogle Scholar
  40. 40.
    Stefanak M., Barnett S.M., Kollar B., Kiss T., Jex I.: Directional correlations in quantum walks with two particles. New J. Phys. 13, 033029 (2011)ADSCrossRefGoogle Scholar
  41. 41.
    Berry S.D., Wang J.B.: Two-particle quantum walks: entanglement and graph isomorphism testing. Phys. Rev. A 83, 042317 (2011)ADSCrossRefGoogle Scholar
  42. 42.
    Romanelli A.: Distribution of chirality in the quantum walk: Markov process and entanglement. Phys. Rev. A 81, 062349 (2011)ADSCrossRefGoogle Scholar
  43. 43.
    Ahlbrecht, A., Alberti, A., Meschede, D., Scholz, V.B., Werner, A.H., Werner, R.F.: Bound Molecules in an Interacting Quantum Walk, arXiv:1105.1051v1 (2011)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Physics DepartmentUniversity College CorkCorkIreland

Personalised recommendations