Skip to main content

Advertisement

SpringerLink
Log in

Quantum walks: a comprehensive review

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete-time quantum walks.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aaronson, S., Ambainis, A.: Quantum search of spatial regions. In: Proceedings of 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 200–209 (2003)

  2. Aaronson S., Ambainis A.: Quantum search of spatial regions. Theory Comput. 1, 47–79 (2005)

    Article  MathSciNet  Google Scholar 

  3. Abal, G., Donangelo, R., Fort, H.: Asymptotic entanglement in the discrete-time quantum walk. In: Annals of the 1st Workshop on Quantum Computation and Information, pp. 189–200 (2006)

  4. Abal G., Donangelo R., Fort H.: Conditional strategies in iterated quantum games. Phys. A 387, 5326–5332 (2008)

    Article  MathSciNet  Google Scholar 

  5. Abal G., Siri R., Romanelli A., Donangelo R.: Quantum walk on the line: entanglement and non-local initial conditions. Phys. Rev. A 73, 042302 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  6. Abdeldayem, H., Frazier, D.O.: Optical computing: need and challenge. Commun. ACM, September (Special Issue: Beyond silicon: new computing paradigms), pp. 60–62 (2007)

  7. Abramsky S.: A structural approach to reversible computation. Theor. Comput. Sci. 347(3), 441–464 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Agliari E.: Trapping of continuous-time quantum walks on Erdős-Rényi graphs. Phys. A Stat. Mech. Appl. 390(11), 1853–1860 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Advanced Research and Development Activity. QIST 2004. a quantum information science and technology roadmap (2004)

  10. Agliari E., Blumen A., Mülken O.: Quantum-walk approach to searching on fractal structures. Phys. Rev. A 82(1), 012305 (2010)

    Article  ADS  Google Scholar 

  11. Agliari E., Blumen A., Mülken O.: Dynamics of continuous-time quantum walks in restricted geometries. J. Phys. A Math. Theor. 41(44), 445301 (2008)

    Article  ADS  Google Scholar 

  12. Agliari E., Mülken O., Blumen A.: Continuous-time quantum walks and trapping. Int. J. Bifurcation and Chaos 20(2), 271–279 (2010)

    Article  ADS  MATH  Google Scholar 

  13. Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of the 33th ACM Symposium on The Theory of Computation (STOC’01) ACM, pp. 50–59 (2001)

  14. Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proceedings of the 35th annual ACM symposium on Theory of computing, pp. 20–29 (2003)

  15. Aharonov D., van Dam W., Kempe J., Landau Z., Lloyd S., Regev O.: Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J. Comput. 37(1), 166–194 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  17. 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)

  18. Ahlbrecht A., Scholz V.B., Werner A.H.: Disordered quantum walks in one lattice dimension. J. Math. Phys. 52(10), 102201 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  19. Ahlbrecht A., Vogts H., Werner A.H., Werner R.F.: Asymptotic evolution of quantum walks with random coin. J. Math. Phys. 52, 042201 (2011)

    Article  ADS  Google Scholar 

  20. Alagić G., Russell A.: Decoherence in quantum walks on the hypercube. Phys. Rev. A 72, 062304 (2005)

    Article  ADS  Google Scholar 

  21. Albertini F., Alessandro D.: Analysis of quantum walks with time-varying coin on d-dimensional lattices. J. Math. Phys. 50, 122106 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  22. Albertini, F., Alessandro, D.: Controllability of quantum walks on graphs. Math. Control Signals Syst. (MCSS), 1–29 (2012). doi:10.1007/s00498-012-0084-0

  23. Allés, B., Gündüc, S., Gündüc, Y.: Maximal entanglement from quantum random walks. Quantum Inf. Process. 1–17 (2011). doi:10.1007/s11128-011-0240-3

  24. Altenkirch, T., Grattage, J.: A functional quantum programming language. In: Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science (LICS 2005), pp. 249–258 (2005)

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

    Article  MATH  Google Scholar 

  26. Ambainis A.: Quantum search algorithms. SIGACT News 35(2), 22–35 (2004)

    Article  Google Scholar 

  27. Ambainis, A.: Quantum walk algorithm for element distinctness. In: Proceedings of 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS’04), pp. 22–31 (2004)

  28. Ambainis, A.: Quantum walk algorithm for element distinctness. In: Proceedings 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 22–31 (2004)

  29. Ambainis, A.: Quantum random walks, a new method for designing quantum algorithms. In: SOFSEM 2008: Theory and Practice of Computer Science, Lecture Notes in Computer Science, vol. 4910, pp. 1–4. Springer, Berlin (2008)

  30. Ambainis, A.: New developments in quantum algorithms. In: 35th International Symposium on Mathematical Foundations of Computer Science, pp. 1–11 (2011)

  31. Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of the 33th ACM Symposium on The Theory of Computation (STOC’01) ACM, pp. 60–69 (2001)

  32. Ambainis, A., Backurs, A., Nahimovs, N., Ozols, R., Rivosh, A.: Search by Quantum Walks on Two-Dimensional Grid Without Amplitude Amplification. arXiv:1112.3337v1 (2011)

  33. Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of 16th ACM-SIAM SODA, pp. 1099–1108 (2005)

  34. Ampadu C.: Limit theorems for quantum walks associated with hadamard matrices. Phys. Rev. A 84(1), 012324 (2011)

    Article  ADS  Google Scholar 

  35. Ampadu, C.: Limit Theorems for the Fibonacci Quantum Walk. arXiv:1108.5198v1 (2011)

  36. Ampadu, C.: Localization of m-Particle Quantum Walks. arXiv:1106.5234v1 (2011)

  37. Ampadu, C.: Localization of Two-Dimensional Five-State Quantum Walks. arXiv:1108.0984v1 (2011)

  38. Ampadu, C.: M-particle Quantum Walks with δ Interaction. arXiv:1105.6076v3 (2011)

  39. Ampadu, C.: On the Recurrence Character of the Hadamard Walk in the Plane. arXiv:1110.0681v1 (2011)

  40. Ampadu C.: On the von Neumann and Shannon entropies for quantum walks on \({\mathbb{Z}^2}\). Int. J. Quantum Inf. 10(2), 1250020 (2012)

    Article  Google Scholar 

  41. Ampadu, C.: Sojourn Times for the One Dimensional Grover Walk. arXiv:1109.2259v1 (2011)

  42. Ampadu, C.: Spectral Analysis of Discrete-Time Quantum Walks in the Quarter Plane. arXiv:1109.3507v1 (2011)

  43. Ampadu, C.: von Neumann Entanglement and Decoherence in Two Dimensional Quantum Walks. arXiv:1110.1917v1 (2011)

  44. Ampadu C.: Brun-type formalism for decoherence in two-dimensional quantum walks. Commun. Theor. Phys. 57(1), 41–55 (2012)

    Article  Google Scholar 

  45. Anderson P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109(5), 1492–1505 (1958)

    Article  ADS  Google Scholar 

  46. Andrade F.M., da Luz M.G.E.: Equivalence between discrete quantum walk models in arbitrary topologies. Phys. Rev. A 80(5), 052301 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  47. Anishchenko, A., Blumen, A., Muelken, O.: Enhancing the spreading of quantum walks on star graphs by additional bonds. Quantum Inf. Process. (2012). doi:10.1007/s11128-012-0376-9

  48. Annabestani M., Abolhasani M.R., Abal G.: Asymptotic entanglement in 2d quantum walks. J. Phys. A Math. Theor. 43(7), 075301 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  49. Annabestani M., Akhtarshenas S.J., Abolhasani M.R.: Decoherence in one-dimensional quantum walk. Phys. Rev. A 81(3), 032321 (2010)

    Article  ADS  Google Scholar 

  50. Annabestani, M., Akhtarshenas, S.J., Abolhassani, M.R.: Tunneling Effects in a One-Dimensional Quantum Walk. arXiv:1004.4352v1 (2010)

  51. Aono, M., Hara, M., Aihara, K.: Amoeba-based neurocomputing with chaotic dynamics. Commun. ACM, September (Special Issue: Beyond silicon: new computing paradigms), 69–72 (2007)

  52. Aspuru-Guzik A., Dutoi A.D., Love P.J., Head-Gordon M.: Simulated quantum computation of molecular energies. Science 309(5741), 1704–1707 (2005)

    Article  ADS  Google Scholar 

  53. Attal, S., Petruccione, F., Sinayskiy, I.: Open Quantum Walks on Graphs. Phys. lett. A (376)18, 1535–1576 (2012)

    Google Scholar 

  54. Bach, E., Borisov, L.: Absorption Probabilities for the Two-Barrier Quantum Walk. arXiv:0901.4349v1 (2009)

  55. Bach E., Coppersmith S., Paz Goldshen M., Joynt R., Watrous J.: One-dimensional quantum walks with absorbing boundaries. J. Comput. Syst. Sci. 69(4), 562–592 (2004)

    Article  MATH  Google Scholar 

  56. Bachman R., Fredette E., Fuller J., Landry M., Opperman M., Tamon C., Tollefson A.: Perfect state transfer on quotient graphs. Quantum Inf. Comput. 12(3&4), 293–313 (2012)

    MathSciNet  Google Scholar 

  57. Bacon, D., Leung, D.: Toward a world with quantum computers. Commun. ACM, September (Special Issue: Beyond silicon: new computing paradigms), pp. 55–59 (2007)

  58. Bacon D., van Dam W.: Recent progress in quantum algorithms. Commun. ACM 53(2), 84–93 (2010)

    Article  Google Scholar 

  59. Banerjee S., Srikanth R., Chandrashekar C.M., Rungta P.: Symmetry-noise interplay in a quantum walk on an n-cycle. Phys. Rev. A 78(5), 052316 (2008)

    Article  ADS  Google Scholar 

  60. Bañuls M.C., Orús R., Latorre J.I., Pérez A., Ruiz-Femenía P.: Simulation of many-qubit quantum computation with matrix product states. Phys. Rev. A 73, 022344 (2006)

    Article  ADS  Google Scholar 

  61. Bañuls M.C., Navarrete C., Pérez A., Roldán E.: Quantum walk with a time-dependent coin. Phys. Rev. A 73, 062304 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  62. Baryshnikov Y., Brady W., Bressler A., Pemantle R.: Two-dimensional quantum random walk. J. Stat. Phys. 142(1), 78–107 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  63. Bednarska M., Grudka A., Kurzyński P., Łuczak T., Wójcik A.: Quantum walks on cycles. Phys. Lett. A 317, 21–25 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. Bednarska M., Grudka A., Kurzyński P., Łuczak T., Wójcik A.: Examples of nonuniform limiting distributions for the quantum walk on even cycles. Int. J. Quantum Inf. 2(4), 453–459 (2004)

    Article  MATH  Google Scholar 

  65. Bender C., Orszag S.: Advanced Mathematical Methods for Scientists and Engineers. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1978)

    Google Scholar 

  66. Benioff P.A.: The computer as a physical system: a microscopic quantum mechanical hamiltonian model of computers as represented by Turing machines. J. Stat. Phys. 22(5), 563–591 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  67. Benioff P.A.: Quantum mechanical hamiltonian models of discrete processes that erase their own histories: Application to Turing machines. Int. J. Theor. Phys. 21, 177–201 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  68. Benioff P.A.: Quantum mechanical hamiltonian models of turing machines. J. Stat. Phys. 3(29), 515–546 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  69. Benioff P.A.: Quantum mechanical models of turing machines that dissipate no energy. Phys. Rev. Lett. 48, 1581–1585 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  70. Benioff, P.A.: Space searches with a quantum robot. In: Lomonaco, S., Brandt, H.E. (eds) Quantum Computation and Quantum Information: A millenium volume. AMS Contemporary Mathematics, vol. 305, pp. 1–12 (2002)

  71. Bennett C.H.: Logical reversibility of computation. IBM J. Res. Dev. 17(6), 525–532 (1973)

    Article  MATH  Google Scholar 

  72. Berg H.C.: Random Walks in Biology. Princeton University Press, Princeton (1993)

    Google Scholar 

  73. Berry S.D., Wang J.B.: Quantum walk-based search and centrality. Phys. Rev. A 82(4), 042333 (2010)

    Article  ADS  Google Scholar 

  74. Berry S.D., Wang J.B.: Two-particle quantum walks: entanglement and graph isomorphism testing. Phys. Rev. A 83(4), 042317 (2011)

    Article  ADS  Google Scholar 

  75. Best A., Kliegl M., Mead-Gluchacki S., Tamon C.: Mixing of quantum walks on generalized hypercubes. Int. J. Quantum Inf. 6(6), 1135–1148 (2008)

    Article  MATH  Google Scholar 

  76. Bettelli S., Calarco T., Serafini L.: Toward an architecture for quantum programming. Eur. Phys. J. At. Mol. Opt. Plasma Phys. 25, 181–200 (2003)

    Google Scholar 

  77. Bleistein N., Handelsman R.: Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York (1975)

    MATH  Google Scholar 

  78. Bouwmeester, D., Ekert, A., Zeilinger, A. (eds): The Physics of Quantum Information. Springer, Berlin (2001)

    Google Scholar 

  79. Brennen G.K., Ellinas D., Kendon V., Pachos J.K., Tsohantjis I., Wang Z.: Anyonic quantum walks. Ann. Phys. 325(3), 664–681 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  80. Bressler, A., Greenwood, T., Pemantle, R., Petkovsek, M.: Quantum random walk on the integer lattice: Examples and phenomena. In: Algorithmic Probability and Combinatorics, Contemporary Mathematics, vol. 520, pp. 41–60. American Mathematical Society (2010)

  81. Bressler, A., Pemantle, R.: Quantum random walks in one dimension via generating functions. In: Proceedings of the 2007 Conference on Analysis of Algorithms, pp. 403–414 (2007)

  82. Briegel H.J., Browne D.E., Dür W., Raussendorf R., Vanden Nest M.: Measurement-based quantum computation. Nat. Phys. 5(1), 19–26 (2009)

    Article  Google Scholar 

  83. 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(15), 153602 (2010)

    Article  ADS  Google Scholar 

  84. Browne D.E.: Efficient classical simulation of the quantum fourier transform. New J. Phys. 9(5), 146 (2007)

    Article  ADS  Google Scholar 

  85. Brun T.A., Carteret H.A., Ambainis A.: Quantum random walks with decoherent coins. Phys. Rev. A 67, 032304 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  86. Brun T.A., Carteret H.A., Ambainis A.: Quantum to classical transition for random walks. Phys. Rev. Lett. 91, 130602 (2003)

    Article  ADS  Google Scholar 

  87. Brun T.A., Carteret H.A., Ambainis A.: Quantum walks driven by many coins. Phys. Rev. A 67, 052317 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  88. Buhrman, H., Špalek, R.: Quantum verification of matrix products. In: Proceedings of the 17th annual ACM-SIAM symposium on Discrete Algorithms, pp. 880–889 (2006)

  89. Cantero M.J., Grünbaum F., Moral L., Velázquez L.: One-dimensional quantum walks with one defect. Rev. Math. Phys. 24(2), 1250002 (2012)

    Article  MathSciNet  Google Scholar 

  90. Cantero M.J., Moral L., Grünbaum F., Velázquez L.: Matrix-valued szegö polynomials and quantum random walks. Commun. Pure Appl. Math. 63(4), 464–507 (2010)

    MATH  Google Scholar 

  91. Caraiman S., Manta V.: Parallel simulation of quantum search. Int. J. Comput. Commun. Control 5, 634–641 (2010)

    Google Scholar 

  92. Carneiro I., Loo M., Xu X., Girerd M., Kendon V., Knight P.L.: Entanglement in coined quantum walks on regular graphs. New J. Phys. 7, 156 (2005)

    Article  ADS  Google Scholar 

  93. Carteret H.A., Ismail M.E.H., Richmond B.: Three routes to the exact asympstotics for the one-dimensional quantum walk. J. Phys. A Math. Gen. 36(33), 8775–8795 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  94. Carteret H.A., Richmond B., Temme N.M.: Evanescence in coined quantum walks. J. Phys. A Math. Gen. 38, 8641–8665 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  95. Chandrashekar C.M.: Generic quantum walk using a coin-embedded shift operator. Phys. Rev. A 78(5), 052309 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  96. Chandrashekar C.M.: Zeno subspace in quantum-walk dynamics. Phys. Rev. A 82(5), 052108 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  97. Chandrashekar, C.M.: Discrete Time Quantum Walk Model for Single and Entangled Particles to Retain Entanglement in Coin Space. arXiv:quant-ph/0609113v4 (2006)

  98. Chandrashekar C.M.: Fractional recurrence in discrete-time quantum walks. Cent. Eur. J. Phys. 8(6), 979–988 (2010)

    Article  Google Scholar 

  99. Chandrashekar, C.M.: Discrete-Time Quantum Walk and its Hamiltonian Form in Different Lattices Along with Temporal, Spatial, and Fluctuating Disordered Operations. arXiv:1103.2704v3 (2011)

  100. Chandrashekar C.M.: Disordered-quantum-walk-induced localization of a bose-einstein condensate. Phys. Rev. A 83(2), 022320 (2011)

    Article  ADS  Google Scholar 

  101. Chandrashekar C.M., Banerjee S.: Parrondo’s games using a discrete-time quantum walk. Phys. Lett. A 375(14), 1553–1558 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  102. Chandrashekar C.M., Banerjee S., Srikanth R.: Relationship between quantum walks and relativistic quantum mechanics. Phys. Rev. A 81(6), 062340 (2010)

    Article  ADS  Google Scholar 

  103. Chandrashekar C.M., Goyal S., Banerjee S.: Entanglement generation in Spatially Separated Systems using quantum walk. Journal of Quantum Information Science 2(2), 15–22 (2012)

    Article  Google Scholar 

  104. Chandrashekar C.M., Srikanth R., Banerjee S.: Symmetries and noise in the quantum walk. Phys. Rev. A 76, 022316 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  105. Chandrashekar C.M., Srikanth R., Laflamme Raymond: Optimizing the discrete quantum walk using a SU(2) coin. Phys. Rev. A 77, 032326 (2007)

    Article  ADS  Google Scholar 

  106. Chase, B.A., Landhal, A.J.: Universal Quantum Walks and Adiabatic Algorithms by 1d Hamiltonians. arXiv:0802.1207 (2008)

  107. Chiang C.-F., Nagal D., Wocjan P.: Efficient circuits for quantum walks. Quantum Inf. Comput. 10(5&6), 420–434 (2010)

    MathSciNet  ADS  MATH  Google Scholar 

  108. Chiang, C.F.: Sensitivity of quantum walks with perturbation. In: Proceedings of the 10th Asian Conference on Quantum Information Science, pp. 209–214 (2011)

  109. Chiang, C.F., Gomez, G.: Hitting time of quantum walks with perturbation. Quantum Inf. Process. 1–12 (2012). doi:10.1007/s11128-012-0368-9

  110. Childs A.: On the relationship between continuous- and discrete-time quantum walk. Commun. Math. Phys. 294(2), 581–603 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  111. Childs A., Eisenberg J.M.: Quantum algorithms for subset finding. Quantum Inf. Comput. 5(7), 593–604 (2005)

    MATH  Google Scholar 

  112. Childs A., Farhi E., Gutmann S.: An example of the difference between quantum and classical random walks. Quantum Inf. Process. 1(1&2), 35–43 (2002)

    Article  MathSciNet  Google Scholar 

  113. Childs A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  114. Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.: Exponential algorithmic speedup by quantum walk. In: Proceedings of the 35th ACM Symposium on The Theory of Computation (STOC’03) ACM, pp. 59–68 (2003)

  115. Childs A.M., Goldstone J.: Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  116. Childs, A.M., Schulman, L.J., Vazirani, U.V.: Quantum algorithms for hidden nonlinear structures. In: Proceedings of 48th IEEE Symposium on Foundations of Computer Science (FOCS 2007), pp. 395–404 (2007)

  117. Childs A.M., Dam W.: Quantum algorithms for algebraic problems. Rev. Mod. Phys. 82, 1–52 (2010)

    Article  ADS  MATH  Google Scholar 

  118. Chisaki K., Hamada M., Konno N., Segawa E.: Limit theorems for discrete-time quantum walks on trees. Interdiscip. Inf. Sci. 15, 423–429 (2009)

    MathSciNet  MATH  Google Scholar 

  119. Chisaki K., Konno N., Segawa E.: Limit theorems for the discrete-time quantum walk on a graph with joined half lines. Quantum Inf. Comput. 12(3&4), 314–333 (2012)

    MathSciNet  Google Scholar 

  120. Chisaki K., Konno N., Segawa E., Shikano Y.: Crossovers induced by discrete-time quantum walks. Quantum Inf. Comput. 11(9&10), 741–760 (2011)

    MathSciNet  MATH  Google Scholar 

  121. Cleve, R., Gavinsky, D., Yonge-Mallo, D.L.: Quantum algorithms for evaluating min-max trees. In: Theory of Quantum Computation, Communication, and Cryptography, pp. 11–15. Springer (2008)

  122. Cohen-Tannoudji C., Diu B., Laloe F.: Quantum Mechanics. Wiley-Interscience, New York (1977)

    Google Scholar 

  123. Coleman R.: Stochastic Processes. George Allen&Unwin, London (1974)

    Google Scholar 

  124. D’Alessandro D., Parlangeli G., Albertini F.: Non-stationary quantum walks on the cycle. J. Phys. A Math. Theor. 40, 14447–14455 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  125. Darázs Z., Kiss T.: Pólya number of the continuous-time quantum walks. Phys. Rev. A 81(6), 062319 (2010)

    Article  ADS  Google Scholar 

  126. de Falco D., Tamascelli D.: Speed and entropy of an interacting continuous time quantum walk. J. Phys. A Math. Gen. 39, 5873–5895 (2006)

    Article  ADS  MATH  Google Scholar 

  127. De Raedt K., Michielsen K., De Raedt H., Trieu B., Arnold G., Richter M., Lippert Th., Watanabe H., Ito N.: Massively parallel quantum computer simulator. Comput. Phys. Commun. 176(2), 121–136 (2007)

    Article  ADS  MATH  Google Scholar 

  128. de Valcárcel G.J., Roldán E., Romanelli A.: Tailoring discrete quantum walk dynamics via extended initial conditions. New J. Phys. 12, 123022 (2010)

    Article  Google Scholar 

  129. Deutsch D.: Quantum theory, the church-turing principle and the universal quantum computer. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 400(1818), 97–117 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  130. Deutsch D., Ekert A., Lupacchini R.: Machines, logic and quantum physics. Bull. Symb. Logic 6(3), 265–283 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  131. Deutsch D., Josza R.: Rapid solutions of problems by quantum computation. Proc. R. Soc. Lond. 439(A), 553–558 (1992)

    ADS  MATH  Google Scholar 

  132. Díaz-Pier S., Venegas-Andraca S.E., Gómez-Muñoz J.L.: Classical simulation of quantum adiabatic algorithms using Mathematica on GPUs. Int. J. Unconv. Comput. 5(7), 315–330 (2011)

    Google Scholar 

  133. Dieks D.: Communication by EPR devices. Phys. Lett. A 92(6), 271–272 (1982)

    Article  ADS  Google Scholar 

  134. Di Franco C., Mc Gettrick M., Machida T., Busch T.: Alternate two-dimensional quantum walk with a single-qubit coin. Phys. Rev. A 84(4), 042337 (2011)

    Article  ADS  Google Scholar 

  135. Di Franco C., Mc Gettrick M., Busch T.: Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin. Phys. Rev. Lett. 106(8), 080502 (2011)

    Article  ADS  Google Scholar 

  136. Dimcovic Z., Rockwell D., Milligan I., Burton R.M., Nguyen T., Kovchegov Y.: Framework for discrete-time quantum walks and a symmetric walk on a binary tree. Phys. Rev. A 84(3), 032311 (2011)

    Article  ADS  Google Scholar 

  137. Di Molfetta, G., Debbasch, F.: Discrete-time quantum walks: continuous limit and symmetries. arXiv:1111.2165v1 (2011)

  138. Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks. The Carus Math. Monographs (28), Mathematical Association of America (1984)

  139. Drezgich M., Hines A.P., Sarovar M., Sastry S.: Complete characterization of mixing time for the continuous quantum walk on the hypercube with markovian decoherence model. Quantum Inf. Comput. 9(9&10), 856–878 (2009)

    MathSciNet  MATH  Google Scholar 

  140. Endo T., Osano S., Toyoshima K., Hirayoshi Y.: Ballistic quantum walk in a discrete one-dimensional system. J. Phys. Soc. Japan 78(6), 064004 (2009)

    Article  ADS  Google Scholar 

  141. Endrejat J., Büttner H.: Entanglement measurement with discrete multiple-coin quantum walks. J. Phys. A Math. Gen. 38(42), 9289 (2005)

    Article  ADS  MATH  Google Scholar 

  142. ERA-Pilot.: Quantum information processing and communication strategic report version 1.4 (2007)

  143. Ermann L., Paz J.P., Saraceno M.: Decoherence induced by a chaotic environment: a quantum walker with a complex coin. Phys. Rev. A 73, 012302 (2006)

    Article  ADS  Google Scholar 

  144. Ezawa M.: Skyrmion burst and multiple quantum walk in thin ferromagnetic films. Phys. Lett. A 375, 3610–3613 (2011)

    Article  ADS  Google Scholar 

  145. Fan S., Feng Z., Xiong S., Yang W.-S.: Convergence of quantum random walks with decoherence. Phys. Rev. A 84(4), 042317 (2011)

    Article  ADS  Google Scholar 

  146. Farhi E., Goldstone J., Gutmann S.: A quantum algorithm for the hamiltonian nand tree. Theory Comput. 4, 169–190 (2008)

    Article  MathSciNet  Google Scholar 

  147. Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum Computation by Adiabatic Evolution. arXiv:quant-ph/0001106 (2000)

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

    Article  MathSciNet  ADS  Google Scholar 

  149. Feinsilver P., Kocik J.: Krawtchouk matrices from classical and quantum walks. Contemp. Math. 287, 83–96 (2002)

    Article  MathSciNet  Google Scholar 

  150. Feldman E., Hillery M.: Scattering theory and discrete-time quantum walks. Phys. Lett. A 324(4), 277–281 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  151. Feldman E., Hillery M.: Modifying quantum walks: a scattering theory approach. J. Phys. A Math. Theor. 40(37), 11343–11359 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  152. Feldman E., Hillery M., Lee H.W., Reitzner D., Zheng H., Bužek V.: Finding structural anomalies in graphs by means of quantum walks. Phys. Rev. A 82(4), 040301 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  153. Feng, T.-L., Zhang, Y.-S., Zhao, G.-M., Liu, S., Guo, G.-C.: Quantum Random Walks of Waves. arXiv:1010.5295v2 (2010)

  154. Fenner, S.A., Zhang, Y.: A note on the classical lower bound for a quantum walk algorithm. quant-ph/0312230 (2003)

  155. Feynman R.P.: Feynman Lectures on Computation. Penguin Books, London (1999)

    Google Scholar 

  156. Feynman R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21(6/7), 467–488 (1982)

    Article  MathSciNet  Google Scholar 

  157. Feynman R.P.: Quantum mechanical computers. Found. Phys. 16(6), 507–531 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  158. Feynman R.P., Leighton R.B., Sands M.: The Feynman Lectures on Physics, vol. III. Addision-Wesley, Reading (1965)

    Google Scholar 

  159. Fjeldsø N., Midtdal J., Ravndal F.: Random walks of a quantum particle on a circle. J. Phys. A Math. Gen. 21, 1633–1647 (1988)

    Article  ADS  Google Scholar 

  160. Fuss, I., White, L., Sherman, P., Naguleswaran, S.: An Analytic Solution for One-Dimensional Quantum Walks. arXiv:0705.0077v1 (2007)

  161. Galindo A., Martin-Delgado M.A.: Information and computation: classical and quantum aspects. Rev. Mod. Phys. 74(2), 347–423 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  162. Gönülol M., Aydiner E., Müstecaplıoğlu ö.E.: Decoherence in two-dimensional quantum random walks with traps. Phys. Rev. A 80(2), 022336 (2009)

    Article  Google Scholar 

  163. Gönülol M., Aydiner E., Shikano Y., Müstecaplıoğlu ö.E.: Survival probability in a one-dimensional quantum walk on a trapped lattice. New J. Phys. 13, 033037 (2011)

    Article  Google Scholar 

  164. Gábris A., Kiss T., Jex I.: Scattering quantum random-walk search with errors. Phys. Rev. A 76, 062315 (2007)

    Article  ADS  Google Scholar 

  165. Gamble J., Friesen M., Zhou D., Joynt R., Coppersmith S.N.: Two-particle quantum walks applied to the graph isomorphism problem. Phys. Rev. A 81(5), 052313 (2010)

    Article  ADS  Google Scholar 

  166. Gay S.: Quantum programming languages: survey and bibliography. Math. Struct. Comput. Sci. 16(4), 581–600 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  167. Ghoshal, D., Lanzagorta, M., Venegas-Andraca, S.E.: A statistical and comparative study of quantum walks under weak measurements and weak values regimes. In: Proceedings (8057) of the SPIE Conference on Defense, Security and Sensing, p. 80570I (2011)

  168. Godoy S., Fujita S.: A quantum random-walk model for tunneling diffusion in a 1d lattice. J. Chem. Phys. 97(7), 5148–5154 (1992)

    Article  ADS  Google Scholar 

  169. Godsil, C.: Average Mixing of Continuous Quantum Walks. arXiv:1103.2578v3 (2011)

  170. Godsil, C., Guo, K.: Quantum Walks on Regular Graphs and Eigenvalues. The Electronic Journal of Combinatorics (18) 1, p. 165 (2011)

  171. Gómez-Muñoz, J.L.: Quantum©, a Mathematica© Add-on for Simulating Quantum Walks and Quantum Algorithms in General. Quantum Information Processing Group, Tecnológico de Monterrey Campus Estado de México. http://homepage.cem.itesm.mx/lgomez/quantum/index.htm (2008)

  172. Goswami S., Sen P., Das A.: Quantum persistence: a random-walk scenario. Phys. Rev. E 81(2), 021121 (2010)

    Article  ADS  Google Scholar 

  173. Gottlieb A.D., Janson S., Scudo P.F.: Convergence of coined quantum walks on \({\mathbb{R}^d}\). Inf. Dim. Anal. Quantum Probab. Rel. Topics 8, 129 (2005)

    Article  MathSciNet  Google Scholar 

  174. Goyal S., Chandrashekar C.M.: Spatial entanglement using a quantum walk on a many-body system. J. Phys. A Math. Theor. 43(23), 235303 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  175. Grady L.: Random walks for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1–17 (2006)

    Article  ADS  Google Scholar 

  176. Grimmett G., Janson S., Scudo P.F.: Weak limits for quantum random walks. Phys. Rev. E 69(2), 02611 (2004)

    Article  Google Scholar 

  177. Grinstread C.M., Snell J.L.: Introduction to Probability. American Mathematical Society, Providence (1997)

    Google Scholar 

  178. Gross D., Nesme V., Vogts H., Werner R.F.: Index theory of one-dimensional quantum walks and cellular automata. Commun. Math. Phys. 310, 419–454 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  179. Grover, L., Rudolph, T.: Creating Superpositions that Correspond to Efficiently Integrable Probability Distributions. arXiv:quant-ph/0208112v1 (2002)

  180. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pp. 212–219 (1996)

  181. Grünbaum, A. Velázquez, L.: The Quantum Walk of F. Riesz. arXiv:1111.6630v1 (2011)

  182. Grünbaum, A., Velázquez, L., Werner, A.H., Werner, R.F.: Recurrence for Discrete Time Unitary Evolutions. arXiv:1202.3903v2 (2012)

  183. Gruska J.: Quantum Computing. McGraw-Hill, New York (1999)

    Google Scholar 

  184. Gudder, S., Sorkin, R.D.: Two-Site Quantum Random Walk. arXiv:1105.0705v1 (2011)

  185. Gudder S.P.: Quantum probability. Academic Press, New York (1988)

    MATH  Google Scholar 

  186. Hamada M., Konno N., Mlotkowski W.: Orthogonal polynomials induced by discrete-time quantum walks in one dimension. Interdiscip. Inf. Sci. 15(3), 367–375 (2009)

    MathSciNet  MATH  Google Scholar 

  187. Hamada, M., Konno, N., Segawa, E.: Relation between coined quantum walks and quantum cellular automata. In: RIMS Kokyuroku, pp. 1–11 (2005)

  188. Hamilton C.S., Gábris A., Jex I., Barnett S.M.: Quantum walk with a four dimensional coin. New J. Phys. 13, 013015 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  189. Hamza, E., Joye, A.: Correlated Markov Quantum Walks. arXiv:1110.4862v2 (2011)

  190. Helmstetter A., Sornette D.: Diffusion of epicenters of earthquake aftershocks, Omori’s law, and generalized continuous-time random walk models. Phys. Rev. E 66(6), 061104 (2002)

    Article  ADS  Google Scholar 

  191. Hillery M., Bergou J., Feldman E.: Quantum walks based on an interferometric analogy. Phys. Rev. A 68, 032314 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  192. Hillery M., Reitzner D., Bužek V.: Searching via walking: how to find a marked clique of a complete graph using quantum walks. Phys. Rev. A 81(6), 062324 (2010)

    Article  ADS  Google Scholar 

  193. Hines A.P., Stamp P.C.E.: Quantum walks, quantum gates, and quantum computers. Phys. Rev. A 75, 062321 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  194. Horn D., Gottlieb A.: Algorithm for data clustering in pattern recognition problems based on quantum mechanics. Phys. Rev. Lett. 88, 18702 (2002)

    Article  ADS  Google Scholar 

  195. Hoyer S., Meyer D.A.: Faster transport with a directed quantum walk. Phys. Rev. A 79(2), 024307 (2009)

    Article  ADS  Google Scholar 

  196. Hull J.C.: Options, Futures and Other Derivatives, 6th edn. Prentice Hall, Upper Saddle River (2005)

    Google Scholar 

  197. Ide Y., Konno N.: Continuous-time quantum walks on the threshold network model. Math. Struct. Comput. Sci. 20(6), 1079–1090 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  198. Ide Y., Konno N., Machida T.: Entanglement for discrete-time quantum walks on the line. Quantum Inf. Comput. 11(9& 10), 855–866 (2011)

    MathSciNet  MATH  Google Scholar 

  199. Ide Y., Konno N., Machida T., Segawa E.: Return probability of one-dimensional discrete-time quantum walks with final-time dependence. Quantum Inf.Comput. 11(9&10), 761–773 (2011)

    MathSciNet  MATH  Google Scholar 

  200. Ide, Y., Konno, N., Segawa, E.: Time averaged distribution of a discrete-time quantum walk on the path. Quantum Information Processing, doi:10.1007/s11128-012-0424-5 (2012)

  201. Iliyasu A.M., Le P.Q., Dong F., Hirota K.: A framework for representing and producing movies on quantum computers. Int. J. Quantum Inf. 9(6), 1459–1497 (2011)

    Article  MATH  Google Scholar 

  202. Iliyasu A.M., Le P.Q., Dong F., Hirota K.: Watermarking and authentication of quantum images based on restricted geometric transformations. Inf. Sci. 186(1), 126–149 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  203. Imre S., Balázs F.: Quantum Computing and Communications: An Engineering Approach. Wiley, New York (2005)

    Google Scholar 

  204. Inui, N., Kasahara, K., Konishi, Y., Konno, N.: Temporal Fluctuations of Continuous-Time Quantum Random Walks on Circles. arXiv:quant-ph/0402062v2 (2004)

  205. Inui N., Konishi Y., Konno N.: Localization of two-dimensional quantum walks. Phys. Rev. A 69, 052323 (2004)

    Article  ADS  Google Scholar 

  206. Inui N., Konishi Y., Konno N., Soshi T.: Fluctuations of quantum random walks on circles. Int. J. Quantum Inf. 3(3), 535–550 (2005)

    Article  MATH  Google Scholar 

  207. Inui N., Konno N.: Localization of multi-state quantum walk in one dimension. Phys. A 353, 133–144 (2005)

    Article  Google Scholar 

  208. Jacobs, B.: Coalgebraic walks, in quantum and turing computation. In: Proceedings of the 14th International Conference on Foundations of Software Science and Computation Structures, Springer LNCS 6604, pp. 12–26 (2009)

  209. Jacobs, B.: Dagger Categories of Tame Relations. arXiv:1101.1077v1 (2011)

  210. Jafarizadeh M.A., Sufiani R., Salimi S., Jafarizadeh S.: Investigation of continuous-time quantum walk by using Krylov subspace-lanczos algorithm. Eur. Phys. J. B 59(2), 199–216 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  211. Jeong H., Paternostro M., Kim M.S.: Simulation of quantum random walks using the interference of a classical field. Phys. Rev. A 69, 012310 (2004)

    Article  ADS  Google Scholar 

  212. Joye, A.: Personal Communication (2012)

  213. Joye, A.: Dynamical localization for d-Dimensional Random Quantum Walks. Quantum Information processing, doi:10.1007/s11128-012-0406-7 (2012)

  214. Joye A.: Random time-dependent quantum walks. Commun. Math. Phys. 307, 65–100 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  215. Joye A., Merkli M.: Dynamical localization of Quantum Walks in Random Environments. J. Stat. Phys. 140, 1025–1053 (2010)

    Article  MathSciNet  Google Scholar 

  216. Kálmán O., Kiss T., Földi P.: Quantum walk on the line with quantum rings. Phys. Rev. B 80(3), 035327 (2009)

    Article  ADS  Google Scholar 

  217. Kargin V.: Bounds for mixing time of quantum walks on finite graphs. J. Phys. A Math. Theor. 43, 335302 (2010)

    Article  MathSciNet  Google Scholar 

  218. Kargin V.: Continuous-time quantum walk on integer lattices and homogeneous trees. J. Stat. Phys. 140, 393–408 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  219. Karski M., Förster L., Choi J.M., Steffen A., Alt W., Meschede D., Widera A.: Quantum walk in position space with single optically trapped atoms. Science 325(5937), 174–177 (2009)

    Article  ADS  Google Scholar 

  220. Kashefi E., Kent A., Vedral V., Banaszek K.: A comparison of quantum oracles. Phys. Rev. A 65, 050304 (2002)

    Article  ADS  Google Scholar 

  221. Kassal I., Aspuru-Guzik A.: Quantum algorithm for molecular properties and geometry optimization. J. Chem. Phys. 131, 224102 (2009)

    Article  ADS  Google Scholar 

  222. Kassal I., Jordan S.P., Love P.J., Mohseni M., Aspuru-Guzik A.: Quantum algorithms for the simulation of chemical dynamics. Proc. Natl. Acad. Sci. 105, 18681 (2008)

    Article  ADS  Google Scholar 

  223. Kassal I., Whitfield J.D., Perdomo-Ortiz A., Yung M.H., Aspuru-Guzik A.: Simulating chemistry using quantum computers. Annu. Rev. Phys. Chem 62, 185–207 (2011)

    Article  ADS  Google Scholar 

  224. Katori M., Fujino S., Konno N.: Quantum walks and orbital states of a weyl particle. Phys. Rev. A 72, 012316 (2005)

    Article  ADS  Google Scholar 

  225. Kempe, J.: PhD thesis: Calcul Quantique—Marches Aléatoires Quantiques et Etude d’Enchevêtrement. école Nationale Supérieure de Télécommunications (2001)

  226. Kempe, J.: Discrete quantum walks hit exponentially faster. In: Proceedings of 7th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM’03), pp. 354–369 (2003)

  227. Kempe J.: Quantum random walks—an introductory overview. Contemp. Phys. 44(4), 307–327 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  228. Kempe, J.: Chapter Quantum algorithms. In: Bruss, D., G. Leuchs (eds.) Lecture Notes on Quantum Information, pp. 87–102. Wiley-VCH (2006)

  229. Kempf A., Portugal R.: Group velocity of discrete-time quantum walks. Phys. Rev. A 79(5), 052317 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  230. Kendon V.: Quantum walks on general graphs. Int. J. Quantum Inf. 4(5), 791–805 (2006)

    Article  MATH  Google Scholar 

  231. Kendon V.: A random walk approach to quantum algorithms. Phil. Trans. R. Soc. A 364(1849), 3407–3422 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  232. Kendon V., Sanders B.C.: Complementarity and quantum walks. Phys. Rev. A 71, 022307 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  233. Kendon, V., Tregenna, B.: Decoherence in a quantum walk on the line. In: Proceedings of QCMC 2002 (2002)

  234. Kendon V., Tregenna B.: Decoherence can be useful in quantum walks. Phys. Rev. A 67, 042315 (2003)

    Article  ADS  Google Scholar 

  235. Kendon, V., Tregenna, B.: Decoherence in discrete quantum walks. In: Selected Lectures from DICE 2002. Lecture Notes in Physics, vol. 633, pp. 253–267 (2003)

  236. Kieferová M., Nagaj D.: Quantum walks on necklaces and mixing. Int. J. Quantum Inf. 10(2), 1250025 (2012)

    Article  Google Scholar 

  237. Kitaev, A.Y., Shen, A.H., Vyhalyi, M.N.: Classical and Quantum Computation. In: Graduate Studies in Mathematics vol. 47. American Mathematical Society (1999)

  238. Kitagawa T., Rudner M.S., Berg E., Demler Eugene E.: Exploring topological phases with quantum walks. Phys. Rev. A 82(3), 033429 (2010)

    Article  ADS  Google Scholar 

  239. Knight P.L., Roldán E., Sipe J.E.: Optical cavity implementations of the quantum walk. Opt. Commun. 227, 147–157 (2003)

    Article  ADS  Google Scholar 

  240. Knight P.L., Roldán E., Sipe J.E.: Quantum walk on the line as an interference phenomenon. Phys. Rev. A 68, 020301 (2003)

    Article  ADS  Google Scholar 

  241. Knight P.L., Roldán E., Sipe J.E.: Propagating quantum walks: the origin of interference structures. J. Mod. Opt. 51(12), 1761–1777 (2004)

    ADS  MATH  Google Scholar 

  242. Kollár B., Štefaň á M., Kiss T., Jex I.: Recurrences in three-state quantum walks on a plane. Phys. Rev. A 82(1), 012303 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  243. Kong, J.: Computation with carbon nanotube devices. Commun. ACM, September (Special Issue: Beyond silicon: new computing paradigms), pp. 40–42 (2007)

  244. Konno N.: Limit theorems and absorption problems for quantum random walks in one dimension. Quantum Inf. Comput. 2, 578–595 (2002)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  246. Konno N.: Symmetry of distribution for the one-dimensional hadamard walk. Interdiscip. Inf. Sci. 10, 11–22 (2004)

    MathSciNet  MATH  Google Scholar 

  247. Konno N.: Limit theorem for continuous-time quantum walk on the line. Phys. Rev. E 72(2), 026113 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  248. Konno N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Japan 57, 1179–1195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  249. Konno N.: A path integral approach for disordered quantum walks in one dimension. Fluctuation Noise Lett. 5(4), 529–537 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  250. Konno N.: Continuous-time quantum walks on trees in quantum probability theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(2), 287–297 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  251. Konno N.: Continuous-time quantum walks on ultrametric spaces. Int. J. Quantum Inf. 4(6), 1023–1036 (2006)

    Article  MATH  Google Scholar 

  252. Konno, N.: ‘Quantum walks’ on quantum potential theory. In: Franz, U., Schuermann, M. (eds.) Lecture Notes in Mathematics, pp. 309–452. Springer (2008)

  253. Konno N.: Limit theorems and absorption problems for one-dimensional correlated random walks. Stoch. Models 25(1), 28–49 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  254. Konno N.: One-dimensional discrete-time quantum walks on random environments. Quantum Inf. Process. 8(5), 387–399 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  255. Konno N.: Localization of an inhomogeneous discrete-time quantum walk on the line. Quantum Inf. Process. 9(3), 405–418 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  256. Konno N.: Quantum walks and elliptic integrals. Math. Struct. Comput. Sci. 20(6), 1091–1098 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  257. Konno, N.: A Note on itô’s Formula for Discrete-Time Quantum Walk. arXiv:1112.4335v1 (2011)

  258. Konno N.: Sojourn times of the hadamard walk in one dimension. Quantum Inf. Process. 11(2), 465–480 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  259. Konno, N., Luczak, T., Segawa, E.: Limit measures of inhomogeneous discrete-time quantum walks in one dimension. Quantum Inf. Process. (2012). doi:10.1007/s11128-011-0353-8

  260. Konno N., Machida T.: Limit theorem for quantum walks with memory. Quantum Inf. Comput. 10(11&12), 1004–1017 (2010)

    MathSciNet  MATH  Google Scholar 

  261. Konno, N., Machida, T., Wakasa, T.: The Heun differential equation and the Gauss differential equation related to quantum walks. arXiv:1109.2662v2 (2011)

  262. Konno N., Mistuda K., Soshi T., Yoo H.J.: Quantum walks and reversible cellular automata. Phys. Lett. A 330(6), 408–417 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  263. Konno N., Namiki T., Soshi T., Sudbury A.: Absorption problems for quantum walks in one dimension. J. Phys. A Math. Gen. 36(1), 241–253 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  264. Konno N., Sato I.: On the relationship between quantum walks and zeta functions. Quantum Inf. Comput. 11(2), 341–349 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  265. Konno N., Segawa E.: Localization of discrete-time quantum walks on a half line via the CGMV method. Quantum Inf. Comput. 11, 485–495 (2011)

    MathSciNet  MATH  Google Scholar 

  266. Konno N., Inui N., Segawa E.: One-dimensional three-state quantum walk. Phys. Rev. E 72, 056112 (2005)

    Article  ADS  Google Scholar 

  267. Koš ík J.: Two models of quantum random walk. Cent. Eur. J. Phys. 4, 556–573 (2003)

    Article  Google Scholar 

  268. Koš ík J., Bužek V.: Scattering model for quantum random walks on hypercube. Phys. Rev. A 71, 012306 (2005)

    Article  ADS  Google Scholar 

  269. Koš ík J., Bužek V., Hillery M.: Quantum walks with random phase shifts. Phys. Rev. A 74, 022310 (2006)

    Article  ADS  Google Scholar 

  270. Kramer B., MacKinnon A.: Localization: theory and experiment. Rep. Prog. Phys. 56, 1469–1564 (1993)

    Article  ADS  Google Scholar 

  271. Krovi, H.: PhD thesis: Symmetry in Quantum Walks. University of Southern California (2007)

  272. Krovi H., Brun T.: Hitting time for quantum walks on the hypercube. Phys. Rev. A 73, 032341 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  273. Krovi H., Brun T.: Quantum walks with infinite hitting times. Phys. Rev. A 74, 042334 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  274. Krovi H., Brun T.A.: Quantum walks on quotient graphs. Phys. Rev. A 75, 062332 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  275. Krovi, H., Magniez, F.: Finding is as easy as detecting for quantum works. Proceedings of the 37th international colloquium conference on Automata, languages and programming, 540–551. Springer-verlag, (2010)

  276. Kurzyński P., Wójcik A.: Discrete-time quantum walk approach to state transfer. Phys. Rev. A 83(6), 062315 (2011)

    Article  ADS  Google Scholar 

  277. Lahini, Y., Verbin, M., Huber, S.D., Bromberg, Y., Pugatch, R., Silberberg, Y.: Quantum Walk of Two Interacting Bosons. arXiv:1105.2273v1 (2011)

  278. Landau,L.D., Lifshitz, E.M.: Statistical Physics, 3rd edn, Part 1: vol. 5 (Course of Theoretical Physics, vol. 5). Butterworth-Heinemann (1980)

  279. Lanzagorta M.: Quantum Radar. Morgan and Claypool, San Rafael (2011)

    Google Scholar 

  280. Lanzagorta M., Uhlmann J.: Quantum Computer Science. Morgan and Claypool, San Rafael (2009)

    Google Scholar 

  281. Lanzagorta M., Uhlmann J.: Quantum algorithmic methods for computational geometry. Math. Struct. Comput. Sci. 20(6), 1117–1125 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  282. Lavička H., Potoček V., Kiss T., Lutz E., Jex I.: Quantum walks with jumps. Eur. Phys. J. D At. Mol. Opt. Plasma Phys. 64(1), 119–129 (2011)

    Google Scholar 

  283. Le P.Q., Doyng F., Hirota K.: A flexible representation of quantum images for polynomial preparation, image compression, and processing operations. Quantum Inf. Process. 10(1), 63–84 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  284. Le P.Q., Iliyasu A.M., Dong F., Hirota K.: Efficient color transformations on quantum images. J. Adv. Comput. Intell. Intell. Inf. 15(6), 698–706 (2011)

    Google Scholar 

  285. Le P.Q., Iliyasu A.M., Doyng F., Hirota K.: Strategies for designing geometric transformations on quantum images. Theor. Comput. Sci. 412(15), 1046–1418 (2011)

    Article  Google Scholar 

  286. Lehman L., Zatloukal V., Brennen G.K., Pachos J.K., Wang Z.: Quantum walks with non-abelian anyons. Phys. Rev. Lett. 106(23), 230404 (2011)

    Article  ADS  Google Scholar 

  287. Leung G., Knott P., Bailey J., Kendon V.: Coined quantum walks on percolation graphs. New J. Phys. 12, 123018 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  288. Linden N., Sharam J.: Inhomogeneous quantum walks. Phys. Rev. A 80(5), 052327 (2009)

    Article  ADS  Google Scholar 

  289. List of QC Simulators. http://www.quantiki.org/wiki/index.php/

  290. Liu, C.: Asymptotic distribution of quantum walks on the line with two entangled coins. Quantum Inf. Process. (2012). doi:10.1007/s11128-012-0361-3

  291. Liu C., Petulante N.: One-dimensional quantum random walks with two entangled coins. Phys. Rev. A 79(3), 032312 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  292. Liu C., Petulante N.: On the von neumann entropy of certain quantum walks subject to decoherence. Math. Struct. Comput. Sci. 20(6), 1099–1115 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  293. Liu C., Petulante N.: Quantum walks on the n-cycle subject to decoherence on the coin degree of freedom. Phys. Rev. E 81(3), 031113 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  294. Liu C., Petulante N.: Asymptotic evolution of quantum walks on the n-cycle subject to decoherence on both the coin and position degrees of freedom. Phys. Rev. A 84(1), 012317 (2011)

    Article  ADS  Google Scholar 

  295. Liu C., Petulante N.: On limiting distributions of quantum markov chains. Int. J. Math. Math. Sci. 2011, 740816 (2011)

    MathSciNet  Google Scholar 

  296. Loepp S., Wootters W.K.: Protecting Information: From Classical Error Correction to Quantum Cryptography. Cambridge University Press, New York (2006)

    Book  MATH  Google Scholar 

  297. López-Acevedo O., Gobron T.: Quantum walks on cayley graphs. J. Phys. A Math. Gen. 39, 585–599 (2006)

    Article  ADS  MATH  Google Scholar 

  298. Lovász, L.: Random walks on graphs: a survey. In: Miklós, D., Sós, V.T., Szönyi, T. (eds.) Combinatorics, Paul Erdős is Eighty, vol. 2. János Bolyai Mathematical Society, Budapest, pp. 353–398 (1996)

  299. Lovász, L., Winkler, P.: Mixing times. In: Aldous, D., Propp, J. (eds.) Microsurveys in Discrete Probability. DIMACS Series in Discrete Math. and theor. Computer Science, pp. 85–133. AMS (1998)

  300. Lovett N., Everitt M., Trevers M., Mosby D., Stockton D., Kendon V.: Spatial search using the discrete time quantum walk. Nat. Comput. 79(1), 1–13 (2010)

    Google Scholar 

  301. Lovett N.B., Cooper S., Everitt M., Trevers M., Kendon V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81(4), 042330 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  302. Lovett, N.B., Everitt, M., Heath, R.M., Kendon, V.: The Quantum Walk Search Algorithm: Factors Affecting Efficiency. arXiv:1110.4366v2 (2011)

  303. Machida, T.: Limit Distribution with a Combination of Density Functions for a 2-State Quantum Walk. arXiv:1010.3481 (2010)

  304. Machida T.: Limit theorems for a localization model of 2-state quantum walks. Int. J. Quantum Inf. 9(3), 863–874 (2011)

    Article  MATH  Google Scholar 

  305. Machida T., Konno N.: Limit theorem for a time-dependent coined quantum walk on the line. Nat. Comput. 2(3), 226–235 (2010)

    Article  ADS  Google Scholar 

  306. Machnes S., Sander U., Glaser S.J.,de Fouquieres P., Gruslys A., Schirmer S., Schulte-Herbrueggen T.: Comparing, optimising and benchmarking quantum control algorithms in a unifying programming framework. Phys. Rev. A 84, 022305 (2011)

    Article  ADS  Google Scholar 

  307. MacKay T.D., Bartlett S.D., Stephenson L.T., Sanders B.C.: Quantum walks in higher dimensions. J. Phys. A (Math. Gen.) 35, 2745–2753 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  308. Magniez F., Nayak A.: Quantum complexity of testing group commutativity. Algorithmica 48(3), 221–232 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  309. Magniez F., Nayak A., Richter P., Santha M.: On the hitting times of quantum versus random walks. Algorithmica 63(1–2), 91–116 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  310. Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: Proceedings of 39th ACM Symposium on Theory of Computing, pp. 575–584 (2007)

  311. Magniez F., Santha M., Szegedy M.: Quantum algorithms for the triangle problem. SIAM J. Comput. 37(2), 413–424 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  312. Maloyer O., Kendon V.: Decoherence vs entanglement in coined quantum walks. New J. Phys. 9, 87 (2007)

    Article  ADS  Google Scholar 

  313. Manouchehri, K., Wang, J.B.: Solid state implementation of quantum random walks on general graphs. In: Proceedings of the 2nd International Workshop on Solid State Quantum Computing and Mini school on Quantum Information Science, pp. 56–61 (2008)

  314. Manouchehri K., Wang J.B.: Quantum random walks without walking. Phys. Rev. A 80(6), 060304 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  315. Margolus N.: Looking at nature as a computer. Int. J. Theor. Phys. 42(2), 309–327 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  316. Marquezino F.L., Portugal R.: The qwalk simulator of quantum walks. Comput. Phys. Commun. 179(5), 359–369 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  317. Marquezino F.L., Portugal R., Abal G.: Mixing times in quantum walks on two-dimensional grids. Phys. Rev. A 82(4), 042341 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  318. Marr, D.W.M., Munakata, T.: Micro/nanofluidic computing. Commun. ACM, September (Special Issue: Beyond silicon: new computing paradigms), pp. 64–68 (2007)

  319. Matjeschk R., Schneider C., Enderlein M., Huber T., Schmitz H., Glueckert J., Schaetz T.: Experimental simulation and limitations of quantum walks with trapped ions. New J. Phys. 14, 035012 (2012)

    Article  ADS  Google Scholar 

  320. Matsuoka L., Kasajima T., Hashimoto M., Yokoyama K.: Numerical study on quantum walks implemented on cascade rotational transitions in a diatomic molecule. J. Korean Phys.Soc. 59(4), 2897–2900 (2011)

    Article  Google Scholar 

  321. Mayer K., Tichy M.C., Minit F., Konrad T., Buchleitner A.: Counting statistics of many-particle quantum walks. Phys. Rev. A 83(6), 062307 (2011)

    Article  ADS  Google Scholar 

  322. Mc Gettrick M.: One dimensional quantum walks with memory. Quantum Inf. Comput. 10(5&6), 509–524 (2010)

    MathSciNet  Google Scholar 

  323. Mermin N.D.: From cbits to qbits: Teaching computer scientists quantum mechanics. Am. J. Phys. 71, 23–30 (2003)

    Article  ADS  Google Scholar 

  324. Mermin N.D.: Quantum Computer Science: An Introduction. Cambridge University Press, New York (2007)

    Book  MATH  Google Scholar 

  325. Messiah A.: Quantum Mechanics. Dover, Mineola (1999)

    Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  327. Meyer D.A., Wallach N.R.: Global entanglement in multiparticle systems. J. Math. Phys 43, 4273–4278 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  328. Miszczak J.A.: Models of quantum computation and quantum programming languages. Bull. Pol. Acad. Sci. Technol. Sci. 59(3), 305–324 (2011)

    Google Scholar 

  329. Miyazaki T., Katori M., Konno N.: Wigner formula of rotation matrices and quantum walks. Phys. Rev. A 76(1), 012332 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  330. Mohseni M., Rebentrost P., Lloyd S., Aspuru-Guzik A.: Environment-assisted quantum walks in energy transfer of photosynthetic complexes. J. Chem. Phys. 129(17), 174106 (2008)

    Article  ADS  Google Scholar 

  331. Montanaro A.: Quantum walks on directed graphs. Quantum Inf. Comput. 7(1), 93–102 (2007)

    MathSciNet  MATH  Google Scholar 

  332. Moore, C., Russell, A.: Quantum walks on the hypercube. In: Proceedings of 6th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM’02), LNCS vol. 2483, pp. 164– 178 (2002)

  333. Mosca M.: Chapter “Quantum algorithms” on Encyclopedia of Complexity and Systems Science, 17th edn, pp. 7088–7118. Springer, Berlin (2009)

    Google Scholar 

  334. Motwani R., Raghavan P.: Randomized Algorithms. Cambridge University Press, New Yotk (1995)

    MATH  Google Scholar 

  335. Müken O., Blumen A.: Continuous-time quantum walks: models for coherent transport on complex networks. Phys. Rep. 502(2&3), 37–87 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  336. Mülken O., Pernice V., Blumen A.: Quantum transport on small-world networks: a continuous-time quantum walk approach. Phys. Rev. E 76, 051125 (2007)

    Article  ADS  Google Scholar 

  337. Mukanata, T. (Guest Editor): Special issue. Beyond silicon: new computing paradigms. Commun. ACM, September (Special Issue: Beyond silicon: new computing paradigms), pp. 30–34 (2007)

  338. Navarrete-Benlloch C., Pérez A., Roldán E.: Nonlinear Optical Galton Board. Phys. Rev. A 75(6), 062333 (2007)

    Article  ADS  Google Scholar 

  339. Nayak, A., Vishwanath, A.: Quantum walk on the line. quant-ph/0010117

  340. Nielsen M., Chuang I.: Quantum Computation and Quantum Information, section 10.5.4, The Gottesman-Knill Theorem. Cambridge University Press, New York (2000)

    Google Scholar 

  341. Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  342. Norris J.R.: Markov Chains. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  343. Nyman P.: A symbolic classical computer language for simulation of quantum algorithms. In: Bruza, P., Sofge, D., Lawless, W., van Rijsbergen, K., Klusch, M. (eds) Quantum Interaction, vol. 5494 of Lecture Notes in Computer Science, pp. 158–173. Springer, Berlin (2009)

    Google Scholar 

  344. Obuse H., Kawakami N.: Topological phases and delocalization of quantum walks in random environments. Phys. Rev. B 84(19), 195139 (2011)

    Article  ADS  Google Scholar 

  345. 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(10), 100602 (2005)

    Article  ADS  Google Scholar 

  346. Oliveira A.C., Portugal R., Donangelo R.: Decoherence in two-dimensional quantum walks. Phys. Rev. A 74(1), 012312 (2006)

    Article  ADS  Google Scholar 

  347. Omar Y., Paunković 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 

  348. Ömer, B.: Quantum Programming in QCL, MSc Thesis. The Technical University of Vienna (2000)

  349. Ömer B.: Classical concepts in quantum programming. Int. J. Theor. Phys. 44, 943–955 (2005)

    Article  ADS  MATH  Google Scholar 

  350. 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 

  351. Papadimitriou C.H.: Computational Complexity. Addison Wesley, Reading (1995)

    Google Scholar 

  352. Paparo, G.D., Martin-Delgado, M.A.: Google in a Quantum Network. Scientific Reports (2)444, 1–12 (2012)

    Google Scholar 

  353. Patel A., Raghunathan K.S., Rahaman Md.A.: Search on a hypercubic lattice using a quantum random walk. ii. d = 2. Phys. Rev. A 82(3), 032331 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  354. Patel A., Raghunathan K.S., Rungta P.: Quantum random walks do not need a coin toss. Phys. Rev. A 71, 032347 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  355. Patel P., Rahaman M.A.: Search on a hypercubic lattice using a quantum random walk. i. d >  2. Phys. Rev. A 82(3), 032330 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  356. Pathak P.K., Agarwal G.S.: Quantum random walk of two photons in separable and entangled states. Phys. Rev. A 75(3), 032351 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  357. Pemantle R., Wilson M.: Asymptotics of multivariate sequences, ii. Multiple points of the singular variety. Combin. Probab. Comput. 13, 735–761 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  358. Perdomo A., Truncik C., Tubert-Brohman I., Rose G., Aspuru-Guzik A.: On the construction of model hamiltonians for adiabatic quantum computation and its application to finding low energy conformations of lattice protein models. Phys. Rev. A 78, 012320 (2008)

    Article  ADS  Google Scholar 

  359. Pérez, A., Romanelli, A.: Spatially Dependent Decoherence and Anomalous Diffusion of Quantum Walks. arXiv:1109.0122v1 (2011)

  360. Peruzzo A., Lobino M., Matthews J.C.F., Matsuda N., Politi A., Poulios K., Zhou X.Q., Lahini Y., Ismail N., Wörhoff K., Bromberg Y., Silberberg Y., Thompson M.G., O’Brien J.L.: Quantum walks of correlated photons. Science 329(5998), 1500–1503 (2010)

    Article  ADS  Google Scholar 

  361. Pólya, G.: Über eine aufgabe der wahrscheinlichkeitstheorie betreffend die irrfahrt im straßennetz. English translation: On an Exercise in Probability Concerning the Random Walk in the Road Network. Math. Ann., vol. 84, pp. 149–160 (1921)

  362. Potoček V., Gábris A., Kiss T., Jex I.: Optimized quantum random-walk search algorithms on the hypercube. Phys. Rev. A 79(1), 012325 (2009)

    Article  ADS  Google Scholar 

  363. Rai A., Agarwal G.S., Perk J.H.H.: Transport and quantum walk of nonclassical light in coupled waveguides. Phys. Rev. A 78(4), 042304 (2008)

    Article  ADS  Google Scholar 

  364. Rantanen, H.: Analyzing the Random-Walk Algorithm for SAT. Master’s thesis, Helsinki University of Technology (2004)

  365. Rao B.R., Srikanth R., Chandrashekar C.M., Banerjee S.: Quantumness of noisy quantum walks: a comparison between measurement-induced disturbance and quantum discord. Phys. Rev. A 6(83), 064302 (2011)

    Article  ADS  Google Scholar 

  366. Regensburger, A., Bersch, C., Hinrichs, B., Onishchukov, G., Schreiber, A., Silberhorn, C., Peschel, U.: Zitterbewegung, Bloch Oscillations and Landau-Zener Tunneling in a Quantum Walk. ArXiv:1104.0105v1 (2011)

  367. Reif, J.H., LaBean, T.H.: Autonomous programmable biomolecular devices using self-assembled DNA nanostructures. Commun. ACM, September (Special Issue: Beyond silicon: new computing paradigms), pp. 46–53 (2007)

  368. Ribeiro P., Milman P., Mosseri R.: Aperiodic quantum random walks. Phys. Rev. Lett. 93, 190503 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  369. Rieffel E., Polak W.: An introduction to quantum computing for non-physicists. ACM Comput. Surv. 32(3), 300–335 (2000)

    Article  Google Scholar 

  370. Rieffel E.G., Polak W.H.: Quantum Computing: A Gentle Introduction. MIT Press, Cambridge (2011)

    MATH  Google Scholar 

  371. Ritcher P.C.: Almost uniform sampling via quantum walks. New J. Phys. 9, 72 (2007)

    Article  ADS  Google Scholar 

  372. Robinett, W., Snider, G.S., Kuekes, P.J., Williams, R.S.: Computing with a trillion crummy components. Commun. ACM, September (Special Issue: Beyond silicon: new computing paradigms), pp. 35–39 (2007)

  373. Rohde, P.: Are Quantum Walks the Saviour of Optical Quantum Computing? http://arxiv.org/abs/1010.4608 (2011)

  374. Rohde P.P., Fedrizzi A., Ralph T.C.: Entanglement dynamics and quasi-periodicity in discrete quantum walks. J. Mod. Opt. 59, 710–720 (2012)

    Article  ADS  Google Scholar 

  375. Rohde P.P., Schreiber A., Štefaňák M., Jex I., Silberhorn Földi 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 

  376. Roldán E., Soriano J.C.: Optical implementability of the two-dimensional quantum walk. J. Mod. Opt. 52, 2649–2657 (2005)

    Article  ADS  MATH  Google Scholar 

  377. Romanelli A.: Measurements in the lévy quantum walk. Phys. Rev. A 76, 054306 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  378. Romanelli A.: Driving quantum-walk spreading with the coin operator. Phys. Rev. A 80(4), 042332 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  379. Romanelli A.: The fibonacci quantum walk and its classical trace map. Phys. A 388(18), 3985–3990 (2009)

    Article  MathSciNet  Google Scholar 

  380. Romanelli A.: Distribution of chirality in the quantum walk: Markov process and entanglement. Phys. Rev. A 81(6), 062349 (2010)

    Article  ADS  Google Scholar 

  381. Romanelli, A.: Thermodynamic behaviour of the Quantum walk. Phys. Rev. A (85), 012319 (2012)

    Google Scholar 

  382. Romanelli A., Hernández G.: Quantum walks: decoherence and coin-flipping games. Phys. A 390(6), 1209–1220 (2011)

    Article  MathSciNet  Google Scholar 

  383. Romanelli A., Sicardi Schifino A.C., Abal G., Donangelo R., Siri R.: Markovian behavior and constrained maximization of the entropy in chaotic quantum systems. Phys. Lett. A 313, 325–329 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  384. Romanelli A., Sicardi Schifino A.C., Siri R., Abal G., Auyuanet A., Donangelo R.: Quantum random walk on the line as a markovian process. Phys. A 338(3–4), 395–405 (2004)

    MathSciNet  Google Scholar 

  385. Romanelli A., Siri R., Abal G., Auyuanet A., Donangelo R.: Decoherence in the quantum walk on the line. Phys. A 347c, 137–152 (2005)

    MathSciNet  ADS  Google Scholar 

  386. Rosmanis A.: Quantum snake walk on graphs. Phys. Rev. A 83(2), 022304 (2011)

    Article  ADS  Google Scholar 

  387. Rudnick J., Gaspari G.: Elements of the Random Walk. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  388. Salimi S.: Continuous-time quantum walks on star graphs. Ann. Phys. 324(6), 1185–1193 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  389. Salimi S.: Continuous-time quantum walks on semi-regular spidernet graphs via quantum probability theory. Quantum Inf. Process. 9(1), 75–91 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  390. Salimi S., Jafarizadeh M.A.: Continuous-time classical and quantum random walk on direct product of cayley graphs. Commun. Theor. Phys. 51(6), 1003–1009 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  391. Salimi S., Radgohar R.: The effect of large decoherence on mixing time in continuous-time quantum walks on long-range interacting cycles. J. Phys. B At. Mol. Opt. Phys 43, 025503 (2010)

    Article  ADS  Google Scholar 

  392. Salimi S., Sorouri A.: Pseudo-hermitian continuous-time quantum walks. J. Phys. A Math. Theor. 43, 275304 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  393. Salimi, S., Yosefjani, R.: Asymptotic Entanglement in 1d Quantum Walks with Time-Dependent Coin. arXiv:1012.4566v1 (2010)

  394. Sansoni L., Sciarrino F., Vallone G., Mataloni P., Crespi A., Ramponi R., Osellame R.: Two-particle bosonic-fermionic quantum walk via 3d integrated photonics. Phys. Rev. Lett. 108(1), 010502 (2012)

    Article  ADS  Google Scholar 

  395. Santha, M.: Quantum walk based search algorithms. In: Proceedings of the 5th Theory and Applications of Models of Computation (TAMC08), Xian, LNCS 4978, pp. 31–46 (2008)

  396. Sato F., Katori M.: Dirac equation with an ultraviolet cutoff and a quantum walk. Phys. Rev. A 81(1), 012314 (2010)

    Article  ADS  Google Scholar 

  397. Sato, M., Kobayashi, N., Katori, M., Konno, N.: Large Qudit Limit of One-Dimensional Quantum Walks. arXiv:0802.1997v1 (2008)

  398. Schöning, U.: A probabilistic algorithm for k-sat and constraint satisfaction problems. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pp. 410–414. IEEE (1999)

  399. Schmitz H., Matjeschk R., Schneider C., Glueckert J., Enderlein M., Huber T., Schaetz T.: Quantum walk of a trapped ion in phase space. Phys. Rev. Lett. 103(9), 1090504 (2009)

    Article  ADS  Google Scholar 

  400. Schreiber A., Cassemiro K.N., Potoček V., Gábris A., Jex I., Silberhorn C.: Decoherence and disorder in quantum walks: From ballistic spread to localization. Phys. Rev. Lett. 106(18), 180403 (2011)

    Article  ADS  Google Scholar 

  401. 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. 104(5), 050502 (2010)

    Article  ADS  Google Scholar 

  402. Segawa, E.: Localization of Quantum Walks Induced by Recurrence Properties of Random Walks. arXiv:1112.4982v1 (2011)

  403. Segawa E., Konno N.: Limit theorems for quantum walks driven by many coins. Int. J. Quantum Inf. 6(6), 1231–1243 (2008)

    Article  MATH  Google Scholar 

  404. Selinger, P.: A brief survey of quantum programming languages. In: Proceedings of the 7th International Symposium on Functional and Logic Programming, Nara, Japan, vol. 2998, pp. 1–6. Springer LNCS (2004)

  405. Selinger P.: Towards a quantum programming language. Math. Struct. Comput. Sci. 14(4), 527–586 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  406. Shenvi N., Kempe J., Whaley R.B.: A quantum random walk search algorithm. Phys. Rev. A 67(5), 052307 (2003)

    Article  ADS  Google Scholar 

  407. Shikano Y., Chisaki K., Segawa E., Konno N.: Emergence of randomness and arrow of time in quantum walks. Phys. Rev. A 81(6), 062129 (2010)

    Article  ADS  Google Scholar 

  408. Shikano Y., Katsura H.: Localization and fractality in inhomogeneous quantum walks with self-duality. Phys. Rev. E 82(3), 031122 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  409. Shor P.W.: Polynomial-time algorithms for prime factorization and discrete algorithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  410. Sipser M.: Introduction to the Theory of Computation. PWS Publishing Co., Boston (2005)

    Google Scholar 

  411. Smith, J.: Cellular Algebras and Graph Invariants Based on Quantum Walks. arXiv:1103.0262v1 (2011)

  412. Solntsev A.S., Sukhorukov A.A., Neshev D.N., Kivshar Y.S.: Spontaneous parametric down-conversion and quantum walks in arrays of quadratic nonlinear waveguides. Phys. Rev. Lett. 108(2), 023601 (2012)

    Article  ADS  Google Scholar 

  413. Somma, R.D., Boixo, S., Barnum, H.: Quantum Simulated Annealing. arXiv:0712.1008 (2007)

  414. Souto Ribeiro P.H., Walborn S.P., Raitz C., Davidovich L., Zagury N.: Quantum random walks and wave-packet reshaping at the single-photon level. Phys. Rev. A 78(1), 2012326 (2008)

    Article  ADS  Google Scholar 

  415. Spitzer F.: Principles of Random Walk, 2nd edn. Springer, Berlin (1976)

    Book  Google Scholar 

  416. Srikanth R., Banerjee S., Chandrashekar C.M.: Quantumness in a decoherent quantum walk using measurement-induced disturbance. Phys. Rev. A 81(6), 062123 (2010)

    Article  ADS  Google Scholar 

  417. Stadler, R.: Molecular, chemical, and organic computing. Commun. ACM, September (Special Issue: Beyond silicon: new computing paradigms), pp. 43–45 (2007)

  418. Stang J.B., Rezakhani A.T., Sanders B.C.: Correlation effects in a discrete quantum random walk. J. Phys. A Math. Theor. 42, 175304 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  419. Štefaňák, M., Kollár, B., Kiss, T., Jex, I.: Full revivals in 2-d quantum walks. Phys. Scripta T140 014035 (2010)

  420. Štefaň ák M., Jex I., Kiss T.: Recurrence and P\'olya number of quantum walks. Phys. Rev. Lett. 100(2), 020501 (2008)

    Article  Google Scholar 

  421. Štefaň ák M., Kiss T., Jex I.: Recurrence properties of unbiased coined quantum walks on infinite $d$ dimensional lattices. Phys. Rev. A. 78(3), 032306 (2008)

    Article  ADS  Google Scholar 

  422. Štefaň ák M., Kiss T., Jex I.: Recurrence of biased coined quantum walks on a line. New J. Phys. 11, 043027 (2009)

    Article  Google Scholar 

  423. Štefaňák M., Barnett S.M., Kollár B., Kiss T., Jex I.: Directional correlations in quantum walks with two particles. New J. Phys. 13, 033029 (2011)

    Article  Google Scholar 

  424. Strauch F.W.: Connecting the discrete and continuous-time quantum walks. Phys. Rev. A 74, 030301 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  425. Strauch F.W.: Reexamination of decoherence in quantum walks on the hypercube. Phys. Rev. A 79(3), 032319 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  426. Sun, B., Le, P.Q., Iliyasu, A.M., Yan, F., Adrián García, J., Dong, F., Hirota, K.: A multi-channel representation for images on quantum computers using the RGBα color space. In: Proceedings of the IEEE 7th International Symposium on Intelligent Signal Processing, pp. 160–165 (2011)

  427. Szegedy, M.: Quantum speed-up of markov chain algorithms. In: Proceedings 45th IEEE Symposium on the Foundations of Computer Science, pp. 32–41 (2004)

  428. Tannor D.J.: Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Books, California (2007)

    Google Scholar 

  429. Travaglione B.C., Milburn G.J.: Implementing the quantum random walk. Phys. Rev. A 65, 032310 (2002)

    Article  ADS  Google Scholar 

  430. Tregenna B., Flanagan W., Maile R., Kendon V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5, 83 (2003)

    Article  ADS  Google Scholar 

  431. Trugenberger C.: Probabilistic quantum memories. Phys. Rev. Lett. 87, 067901 (2001)

    Article  ADS  Google Scholar 

  432. Trugenberger C.: Phase transitions in quantum pattern recognition. Phys. Rev. Lett. 89, 277903 (2002)

    Article  ADS  Google Scholar 

  433. Trugenberger C.: Quantum pattern recognition. Quantum Inf. Process. 1(6), 471–493 (2002)

    Article  MathSciNet  Google Scholar 

  434. Tsomokos D.: Quantum walks on complex networks with connection instabilities and community structure. Phys. Rev. A 83(5), 052315 (2011)

    Article  ADS  Google Scholar 

  435. Tulsi A.: Faster quantum-walk algorithm for the two-dimensional spatial search. Phys. Rev. A 78(1), 012310 (2008)

    Article  ADS  Google Scholar 

  436. Underwood M.S., Feder D.L.: Universal quantum computation by discontinuous quantum walk. Phys. Rev. A 82(4), 042304 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  437. van Dam, W.: Quantum Cellular Automata. MSc thesis, University of Nijmegen, The Netherlands (1996)

  438. van Hoogdalem K.A., Blaauboer M.: Implementation of the quantum-walk step operator in lateral quantum dots. Phys. Rev. B 80(12), 125309 (2009)

    Article  ADS  Google Scholar 

  439. Varbanov M., Krovi H., Brun T.A.: Hitting time for the continuous quantum walk. Phys. Rev. A 78(2), 022324 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  440. Vedral V.: Introduction to Quantum Information Science. Oxford University Press, New York (2006)

    Book  MATH  Google Scholar 

  441. Venegas-Andraca, S.E.: DPhil thesis: Discrete Quantum Walks and Quantum Image Processing. Centre for Quantum Computation, University of Oxford (2006)

  442. Venegas-Andraca S.E.: Quantum Walks for Computer Scientists. Morgan and Claypool, San Rafael (2008)

    Google Scholar 

  443. Venegas-Andraca S.E., Ball J.L.: Processing images in entangled quantum systems. Quantum Inf. Process. 9(1), 1–11 (2010)

    Article  MathSciNet  Google Scholar 

  444. Venegas-Andraca S.E., Ball J.L., Burnett K., Bose S.: Quantum walks with entangled coins. New J. Phys. 7, 221 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  445. Venegas-Andraca, S.E., Bose, S.: Quantum computation and image processing: new trends in artificial intelligence. In: Proceedings of the International Conference on Artificial Intelligence IJCAI-03, pp. 1563–1564 (2003)

  446. Venegas-Andraca, S.E., Bose, S.: Storing, processing and retrieving an image using quantum mechanics. In: Proceedings of the SPIE Conference Quantum Information and Computation, pp. 137–147 (2003)

  447. Venegas-Andraca, S.E., Bose, S.: Quantum Walk-Based Generation of Entanglement Between Two Walkers. arXiv:0901.3946v1 (2009)

  448. Viamontes G.F., Markov I., Hayes J.P.: Improving gate-level simulation of quantum circuits. Quantum Inf. Process. 2, 347–380 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  449. Villagra, M., Nakanishi, M., Yamashita, S., Nakashima, Y.: Quantum walks on the line with phase parameters. In: Proceedings of the 10th Asian Conference on Quantum Information Science (AQIS’10) (2010)

  450. Ward S.N.: Earthquake simulation by restricted random walks. Bull. Seismol. Soc. Am. 94(6), 2079–2089 (2004)

    Article  Google Scholar 

  451. Watabe K., Kobayashi N., Katori M., Konno N.: Limit distributions of two-dimensional quantum walks. Phys. Rev. A 77, 062331 (2008)

    Article  ADS  Google Scholar 

  452. Watrous J.: Quantum simulations of classical random walks and undirected graph connectivity. J. Comput. Syst. Sci. 62(2), 376–391 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  453. Whitfield J.D., Rodríguez-Rosario C.A., Aspuru-Guzik A.: Quantum stochastic walks: a generalization of classical random walks and quantum walks. Phys. Rev. A 81(2), 022323 (2010)

    Article  ADS  Google Scholar 

  454. Williams C.P.: Explorations in Quantum Computing. Springer, Berlin (2011)

    Book  Google Scholar 

  455. Witthaut D.: Quantum walks and quantum simulations with bloch-oscillating spinor atoms. Phys. Rev. A 82(3), 033602 (2010)

    Article  ADS  Google Scholar 

  456. Woess W.: Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics (138). Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  457. Wootters W.K., Zurek W.H.: Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of bohr’s principle. Phys. Rev. D 19, 473–484 (1979)

    Article  ADS  Google Scholar 

  458. Wootters W.K., Zurekh W.H.: A single quantum state cannot be cloned. Nature 299, 802–803 (1982)

    Article  ADS  Google Scholar 

  459. Xu X.-P.: Exact analytical results for quantum walks on star graphs. J. Phys. A Math. Theor. 42, 115205 (2009)

    Article  ADS  Google Scholar 

  460. Xu X.P.: Continuous-time quantum walks on one-dimensional regular networks. Phys. Rev. E 77(6), 061127 (2008)

    Article  ADS  Google Scholar 

  461. Xu X.P.: Discrete-time quantum walks on one-dimensional lattices. Eur. Phys. Lett. B 77, 479–488 (2010)

    Article  ADS  Google Scholar 

  462. Xu X.P., Liu F.: Continuous-time quantum walks on erdös rényi networks. Phys. Lett. A 372(45), 6727–6732 (2008)

    Article  ADS  MATH  Google Scholar 

  463. Xue P., Sanders B.C.: Quantum quincunx for walk on circles in phase space with indirect coin ip. New J. Phys. 10, 053025 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  464. Xue P., Sanders B.C.: Two quantum walkers sharing coins. Phys. Rev. A 85, 022307 (2012)

    Article  ADS  Google Scholar 

  465. Xue P., Sanders B.C., Blais A., Lalumière K.: Quantum walks on circles in phase space via superconducting circuit quantum electrodynamics. Phys. Rev. A 78(4), 042334 (2008)

    Article  ADS  Google Scholar 

  466. Xue P., Sanders C.B., Leibfried D.: Quantum walk on a line for a trapped ion. Phys. Rev. Lett. 103(18), 183602 (2009)

    Article  ADS  Google Scholar 

  467. Yamasaki T., Kobayashi H., Imai H.: Analysis of absorbing times of quantum walks. Phys. Rev. A 68, 012302 (2003)

    Article  ADS  Google Scholar 

  468. Yang W-S., Liu C., Zhang K.: A path integral formula with applications to quantum random walks in Z d. J. Phys. A Math. Theor. 40, 8487–8516 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  469. Zähringer 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(10), 100503 (2010)

    Article  Google Scholar 

  470. Zhang K.: Limiting distribution of decoherent quantum random walks. Phys. Rev. A 77(6), 062302 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  471. Zhang P., Liu B.H., Liu R.F., Li H.R., Li F.L., Guo G.C.: Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons. Phys. Rev. A 81(5), 052322 (2010)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tecnológico de Monterrey Campus Estado de México and Texia, SA de CV, Atizapań de Zaragoza, Estado de México, México

    Salvador Elías Venegas-Andraca

Authors
  1. Salvador Elías Venegas-Andraca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Salvador Elías Venegas-Andraca.

Additional information

To my dear friend Carlos Fuentes, In Memoriam.

To my beloved daughter Renata, welcome to my life and to Planet Earth!

Rights and permissions

Reprints and permissions

About this article

Cite this article

Venegas-Andraca, S.E. Quantum walks: a comprehensive review. Quantum Inf Process 11, 1015–1106 (2012). https://doi.org/10.1007/s11128-012-0432-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-012-0432-5

Keywords

Search

Navigation