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

Quantum Information Processing volume 11, pages 1287–1299 (2012)Cite this article

Abstract

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.

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References

  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)]

  2. Feynman R.P., Hibbs A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)

    MATH  Google Scholar 

  3. Aharonov Y., Davidovich L., Zagury N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)

    Article  ADS  Google Scholar 

  4. Meyer D.A.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85, 551–574 (1996)

    Article  ADS  MATH  Google Scholar 

  5. Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  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)

  7. Nayak, A., Vishwanath, A.: Quantum Walk on the Line, DIMACS Technical Report, No. 2000-43 (2001); arXiv:quant-ph/0010117

  8. Ambainis A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1(4), 507–518 (2003)

    Article  MATH  Google Scholar 

  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)

  10. Shenvi N., Kempe J., Birgitta Whaley K.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)

    Article  ADS  Google Scholar 

  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)

  12. Chandrashekar C.M., Laflamme R.: Quantum phase transition using quantum walks in an optical lattice. Phys. Rev. A 78, 022314 (2008)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  14. Engel G.S. et al.: Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature 446, 782–786 (2007)

    Article  ADS  Google Scholar 

  15. Mohseni M., Rebentrost P., Lloyd S., Aspuru-Guzik A.: Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys. 129, 174106 (2008)

    Article  ADS  Google Scholar 

  16. Chandrashekar, C.M., Goyal, S.K., Banerjee, S.: Entanglement generation in spatially separated systems using quantum walk arXiv:1005.3785 (2010)

  17. Kitagawa T., Rudner M.S., Berg E., Demler E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  28. Konno N.: Quantum random walks in one dimension. Quantum Inf. Process. 1(5), 345–354 (2002)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  35. Jaksch D.: Optical lattices, ultracold atoms and quantum information processing. Contemp. Phys. 45(5), 367–381 (2004)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  38. Omar Y., Paunkovic N., Sheridan L., Bose S.: Quantum walk on a line with two entangled particles. Phys. Rev. A 74, 042304 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  41. Berry S.D., Wang J.B.: Two-particle quantum walks: entanglement and graph isomorphism testing. Phys. Rev. A 83, 042317 (2011)

    Article  ADS  Google Scholar 

  42. Romanelli A.: Distribution of chirality in the quantum walk: Markov process and entanglement. Phys. Rev. A 81, 062349 (2011)

    Article  ADS  Google Scholar 

  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)

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Authors and Affiliations

  1. Physics Department, University College Cork, Cork, Ireland

    C. M. Chandrashekar & Th. Busch

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  1. C. M. Chandrashekar
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  2. Th. Busch
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Correspondence to C. M. Chandrashekar.

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Chandrashekar, C.M., Busch, T. Quantum walk on distinguishable non-interacting many-particles and indistinguishable two-particle. Quantum Inf Process 11, 1287–1299 (2012). https://doi.org/10.1007/s11128-012-0387-6

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  • DOI: https://doi.org/10.1007/s11128-012-0387-6

Keywords

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