Theoretical Framework

  • Kia Manouchehri
  • Jingbo Wang
Part of the Quantum Science and Technology book series (QST)


There are two broad classes of quantum walks, one in discrete-time and the other in continuous-time, each of which have independently emerged out of the study of unrelated physical problems. Here we present the theoretical background for both classes and briefly discuss some of their properties.


Position State Quantum Walk Translation Operator Quotient Graph Coin Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Abal, G., Donangelo, R., Fort, H.: Annals of the 1st Workshop on Quantum Computation and Information, pp. 189–200 (2006a). arXiv:quant-ph/0709.3279v1 Google Scholar
  2. Abal, G., Donangelo, R., Romanelli, A., Siri, R.: Physica A 371, 1 (2006b)MathSciNetADSCrossRefGoogle Scholar
  3. Abal, G., Siri, R., Romanelli, A., Donangelo, R.: Phys. Rev. A 73, 042302 (2006c)MathSciNetADSCrossRefGoogle Scholar
  4. Abala, G., Donangeloa, R., Severoa, F., Siria, R.: Physica A 387, 335 (2008)MathSciNetADSCrossRefGoogle Scholar
  5. Adamczak, W., Peter, K.A., Tamon, H.C.: (2003). arXiv:quant-ph/0308073 Google Scholar
  6. Agarwal, G.S., Pathak, P.K.: Phys. Rev. A 72, 033815 (2005)ADSCrossRefGoogle Scholar
  7. Aharonov, Y., Davidovich, L., Zagury, N.: Phys. Rev. A 48, 1687 (1993)ADSCrossRefGoogle Scholar
  8. Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: STOC’01: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 50–59. ACM (2001). arXiv:quant-ph/0012090 Google Scholar
  9. Alagić, G., Russell, A.: Phys. Rev. A 72, 062304 (2005)ADSCrossRefGoogle Scholar
  10. Ambainis, A.: Int. J. Quantum Inf. 1, 507 (2003)zbMATHCrossRefGoogle Scholar
  11. Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: STOC’01: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 50–59. ACM (2001)Google Scholar
  12. Anderson, P.W.: Phys. Rev. 109, 1492 (1958)ADSCrossRefGoogle Scholar
  13. Aslangul, C.: J. Phys. A 38, 1 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. Bach, E., Coppersmith, S., Goldschen, M.P., Joynt, R., Watrous, J.: J. Comput. Syst. Sci. 69, 562 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  15. Berry, S.D., Wang, J.B.: Phys. Rev. A 83, 042317 (2011)ADSCrossRefGoogle Scholar
  16. Blanchard, P., Hongler, M.-O.: Phys. Rev. Lett. 92, 120601 (2004)MathSciNetADSCrossRefGoogle Scholar
  17. Broome, M.A., Fedrizzi, A., Lanyon, B.P., Kassal, I., Aspuru-Guzik, A., White, A.G.: Phys. Rev. Lett. 104, 153602 (2010)ADSCrossRefGoogle Scholar
  18. Brun, T.A., Carteret, H.A., Ambainis, A.: Phys. Rev. Lett. 91, 130602 (2003a)ADSCrossRefGoogle Scholar
  19. Brun, T.A., Carteret, H.A., Ambainis, A.: Phys. Rev. A 67, 052317 (2003b)MathSciNetADSCrossRefGoogle Scholar
  20. Brun, T.A., Carteret, H.A., Ambainis, A.: Phys. Rev. A 67, 032304 (2003c)MathSciNetADSCrossRefGoogle Scholar
  21. Carneiro, I., Loo, M., Xu, X., Girerd, M., Kendon, V., Knight, P.L.: New J. Phys. 7, 156 (2005)ADSCrossRefGoogle Scholar
  22. Childs, A.M.: Commun. Math. Phys. 294, 581 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. Childs, A., Goldstone, J.: Phys. Rev. A 70, 022314 (2004b)MathSciNetADSCrossRefGoogle Scholar
  24. Côté, R., Russell, A., Eyler, E.E., Gould, P.L.: New J. Phys. 8, 156 (2006)CrossRefGoogle Scholar
  25. Du, J., Li, H., Xu, X., Shi, M., Wu, J., Zhou, X., Han, R.: Phys. Rev. A 67, 042316 (2003)ADSCrossRefGoogle Scholar
  26. Dur, W., Raussendorf, R., Kendon, V.M., Briegel, H.J.: Phys. Rev. A 66, 052319 (2002)ADSCrossRefGoogle Scholar
  27. Farhi, E., Gutmann, S.: Phys. Rev. A 58, 915 (1998)MathSciNetADSCrossRefGoogle Scholar
  28. Hillery, M., Bergou, J., Feldman, E.: Phys. Rev. A 68, 032314 (2003)MathSciNetADSCrossRefGoogle Scholar
  29. Keating, J.P., Linden, N., Matthews, J.C.F., Winter, A.: Phys. Rev. A 76, 012315 (2007)ADSCrossRefGoogle Scholar
  30. Kempe, J.: RANDOM’03: Proceedings of the 7th International Workshop on Randomization and Approximation Techniques in Computer Science. Lecture Notes in Computer Science, vol. 2764, pp. 354–369. Springer (2003a). arXiv:quant-ph/0205083 Google Scholar
  31. Kempe, J.: Contemp. Phys. 44, 307 (2003b)ADSCrossRefGoogle Scholar
  32. Kendon, V.: Int. J. Quantum Inf. 4, 791 (2006a)zbMATHCrossRefGoogle Scholar
  33. Kendon, V.: Math. Struct. Comput. Sci. 17, 1169 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  34. Kendon, V., Sanders, B.C.: Phys. Rev. A 71, 022307 (2005)MathSciNetADSCrossRefGoogle Scholar
  35. Kendon, V., Tregenna, B.: QCMC’02: Proceedings of the 6th International Conference on Quantum Communication, Measurement and Computing, p. 463. Rinton Press, (2002). arXiv:quant-ph/0210047 Google Scholar
  36. Kendon, V., Tregenna, B.: Phys. Rev. A 67, 042315 (2003)ADSCrossRefGoogle Scholar
  37. Kendona, V., Maloyera, O.: Theor. Comput. Sci. 394, 187 (2008)CrossRefGoogle Scholar
  38. Knight, P.L., Roldán, E., Sipe, J.E.: Opt. Commun. 227, 147 (2003a)ADSCrossRefGoogle Scholar
  39. Knight, P.L., Roldán, E., Sipe, J.E.: Phys. Rev. A 68, 020301 (2003b)ADSCrossRefGoogle Scholar
  40. Konno, N.: Phys. Rev. E 72, 26113 (2005)MathSciNetADSCrossRefGoogle Scholar
  41. Konno, N., Namiki, T., Soshi, T.: Interdiscip. Inf. Sci. 10, 11 (2004)MathSciNetzbMATHGoogle Scholar
  42. Košík, J., Bužek, V., Hillery, M.: Phys. Rev. A 74, 022310 (2006)ADSCrossRefGoogle Scholar
  43. Krovi, H., Brun, T.A.: Phys. Rev. A 73, 032341 (2006a)MathSciNetADSCrossRefGoogle Scholar
  44. Krovi, H., Brun, T.A.: Phys. Rev. A 74, 042334 (2006b)MathSciNetADSCrossRefGoogle Scholar
  45. Krovi, H., Brun, T.A.: Phys. Rev. A 75, 062332 (2007)MathSciNetADSCrossRefGoogle Scholar
  46. López, C.C., Paz, J.P.: Phys. Rev. A 68, 052305 (2003)ADSCrossRefGoogle Scholar
  47. Maloyer, O., Kendon, V.: New J. Phys. 9, 87 (2007)ADSCrossRefGoogle Scholar
  48. Manouchehri, K., Wang, J.B.: J. Phys. A 40, 13773 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  49. Marquezino, F.L., Portugal, R.: Phys. Rev. A 77, 042312 (2008)MathSciNetADSCrossRefGoogle Scholar
  50. Meyer, D.A.: J. Stat. Phys. 85, 551 (1996)ADSzbMATHCrossRefGoogle Scholar
  51. Montanaro, A.: Quantum Inf. Comput. 7, 93 (2007)MathSciNetzbMATHGoogle Scholar
  52. Moore, C., Russell, A.: RANDOM’02: Proceedings of the 6th International Workshop Randomization and Approximation in Computer Science. Lecture Notes in Computer Science, vol. 2483, pp. 164–178. Springer (2002). arXiv:quant-ph/0104137 Google Scholar
  53. Mülken, O., Blumen, A.: Phys. Rep. 502, 37 (2011)MathSciNetADSCrossRefGoogle Scholar
  54. Nayak, A., Vishwan, A.: DIMACS Technical Report 43 (2000)Google Scholar
  55. Oliveira, A.C., Portugal, R., Donangelo, R.: Phys. Rev. A 74, 012312 (2006)ADSCrossRefGoogle Scholar
  56. Omar, Y., Paunković, N., Bose, L.S.S.: Phys. Rev. A 74, 042304 (2006)MathSciNetADSCrossRefGoogle Scholar
  57. Patel, A., Raghunathan, K.S., Rungta, P.: Phys. Rev. A 71, 032347 (2005)MathSciNetADSCrossRefGoogle Scholar
  58. Pathak, P.K., Agarwal, G.S.: Phys. Rev. A 75, 032351 (2007)MathSciNetADSCrossRefGoogle Scholar
  59. Ribeiro, P., Milman, P., Mosseri, R.: Phys. Rev. Lett. 93, 190503 (2004)MathSciNetADSCrossRefGoogle Scholar
  60. Richter, P.C.: Phys. Rev. A 76, 042306 (2007b)ADSCrossRefGoogle Scholar
  61. Rohde, P.P., Schreiber, A., Štefaňǎk, M., Jex, I., Silberhorn, C.: New J. Phys. 13, 013001 (2011)ADSCrossRefGoogle Scholar
  62. Romanelli, A., Siri, R., Abal, G., Auyuanet, A., Donangelo, R.: Physica A 347, 137 (2005)MathSciNetADSCrossRefGoogle Scholar
  63. Sanders, B.C., Bartlett, S.D.: Phys. Rev. A 67, 042305 (2003)ADSCrossRefGoogle Scholar
  64. Schreiber, A., Cassemiro, K.N., Potoček, V., Gábris, A., Jex, I., Silberhorn, C.: Phys. Rev. Lett. 106, 180403 (2011)ADSCrossRefGoogle Scholar
  65. Shapira, D., Biham, O., Bracken, A.J., Hackett, M.: Phys. Rev. A 68, 062315 (2003)MathSciNetADSCrossRefGoogle Scholar
  66. Shenvi, N., Kempe, J., Whaley, K.B.: Phys. Rev. A 67, 052307 (2003)ADSCrossRefGoogle Scholar
  67. Solenov, D., Fedichkin, L.: Phys. Rev. A 73, 012313 (2006a)MathSciNetADSCrossRefGoogle Scholar
  68. Solenov, D., Fedichkin, L.: Phys. Rev. A 73, 012308 (2006b)MathSciNetADSCrossRefGoogle Scholar
  69. Stefanak, M., Kiss, T., Jex, I., Mohring, B.: J. Phys. A 39, 14965 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  70. Strauch, F.W.: Phys. Rev. A 74, 030301 (2006a)MathSciNetADSCrossRefGoogle Scholar
  71. Townsend, J.S.: A Modern Approach to Quantum Mechanics. University Science Books, Sausalito (2000a)Google Scholar
  72. Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: New J. Phys. 5, 83 (2003)ADSCrossRefGoogle Scholar
  73. Venegas-Andraca, S.E., Bose, S.: (2009). arXiv:0901.3946 Google Scholar
  74. Venegas-Andraca, S.E., Ball, J.L., Burnett, K., Bose, S.: New J. Phys. 7, 221 (2005)MathSciNetADSCrossRefGoogle Scholar
  75. Watrous, J.: J. Comput. Syst. Sci. 62, 376 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  76. Xu, X.-P.: Phys. Rev. E 77, 061127 (2008)ADSCrossRefGoogle Scholar
  77. Yin, Y., Katsanos, D.E., Evangelou, S.N.: Phys. Rev. A 77, 022302 (2008)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Kia Manouchehri
    • 1
  • Jingbo Wang
    • 1
  1. 1.School of PhysicsThe University of Western AustraliaPerthAustralia

Personalised recommendations