Quantum walks and search algorithms

Quantum Walks and Search Algorithms

Volumn , Issue , 2013, Pages 1-222

  Portugal, Renato ^a  

^a LABORATÓRIO NACIONAL DE COMPUTAÇÃO CIENTÍFICA   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85028513137     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-1-4614-6336-8     Document Type: Book

References (83)

1. Aaronson, S., Ambainis, A.: Quantum search of spatial regions. Theory of Computing, 1, 47-79 (2003)
2. Abal, G., Donangelo, R., Marquezino, F.L., Portugal, R.: Spatial search on a honeycomb network. Math. Struct. Comput. Sci. 20(Special Issue 06), 999-1009 (2010)
3. Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of 33th STOC, pp. 50-59. ACM, New York (2001)
4. Aharonov, D.: Quantum computation -a review. In: Stauffer, D. (ed.) Annual Review of Computational Physics, vol. VI, pp. 1-77. World Scientific, Singapore (1998)
5. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48(2), 1687-1690 (1993)
6. Aldous, D.J., Fill, J.A.: Reversible Markov Chains and Random Walks on Graphs. Book in preparation, http://www.stat.berkeley.edu/aldous/RWG/book.html (2002)
7. Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of 33th STOC, pp. 60-69. ACM, New York (2001)
8. Ambainis, A., Backurs, A., Nahimovs, N., Ozols, R., Rivosh, A.: Search by quantum walks on two-dimensional grid without amplitude amplification. arxiv:1112.3337 (2011)
9. Ambainis, A.: Quantum walk algorithm for element distinctness. In: FOCS '04: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 22-31. IEEE Computer Society, Washington, DC (2004)
10. Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1099-1108. SIAM, Philadelphia (2005)
11. Apostol, T.M.: Calculus, vol. 1: One-Variable Calculus with an Introduction to Linear Algebra. Wiley, New York (1967)
12. Axler, S.: Linear Algebra Done Right. Springer, New York (1997)
13. Bednarska, M., Grudka, A., Kurzynski, P., Luczak, T., Wójcik, A.: Quantum walks on cycles. Phys. Lett. A 317(1-2), 21-25 (2003)
14. Bednarska, M., Grudka, A., Kurzynski, P., Luczak, T., Wójcik, A.: Examples of non-uniform limiting distributions for the quantum walk on even cycles. Int. J. Quant. Inform. 2(4), 453-459 (2004)
15. Benioff, P.: Space searches with a quantum robot. (ed.) Samuel J. Lomonaco, Jr. and Howard D. Brandt Contemporary Mathematics, AMS, as a special session about Quantum Computation and Information, vol. 305, pp. 1-12. Washington, D.C (2002)
16. Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.V.: Strengths and weaknesses of quantum computing. SIAM J. Comput. 26(5), 1510-1523 (1997)
17. Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Forstschritte Der Physik 4, 820-831 (1998)
18. Brassard, G., Høyer, P.,Mosca,M., Tapp, A.: Quantum amplitude amplification and estimation. Quant. Comput. Quant. Inform. Sci., Comtemporary Mathematics 305, 53-74, (2002), quantph/ 0005055
19. Carteret, H.A., Ismail, M.E.H., Richmond, B.: Three routes to the exact asymptotics for the one-dimensional quantum walk. J. Phys A: Math. General 36(33), 8775-8795 (2003)
20. Childs, A.: On the relationship between continuous-and discrete-time quantum walk. Commun. Math. Phys. 294, 581-603 (2010)
21. Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)
22. Childs, A.M., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. Quant. Informa. Process. 1(1), 35-43 (2002)
23. Diu, B., Cohen-Tannoudji, C., Laloe, F.: Quantum Mechanics. Wiley-Interscience, New York (2006)
24. Cover, T.M., Thomas, J.: Elements of Information Theory. Wiley, New York (1991)
25. d'Espagnat, B.: Conceptual Foundations of Quantum Mechanics. Westview Press, Boulder (1999)
26. Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915-928 (1998)
27. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968)
28. Forets, M., Abal, G., Donangelo, R., Portugal, R.: Spatial quantum search in a triangular network. Math. Struct. Comput. Sci. 22(03), 521-531 (2012)
29. Gould, H.W.: Combinatorial Identities. Morgantown Printing and Binding Co., Morgantown (1972)
30. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley Professional, Reading (1994)
31. Griffiths, D.: Introduction to Quantum Mechanics, 2nd edn. Benjamin Cummings, Menlo Park (2005)
32. Grover, L.K.: Quantum computers can search arbitrarily large databases by a single query. Phys. Rev. Lett. 79(23), 4709-4712 (1997)
33. Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79(2), 325-328 (1997)
34. Grover, L.K.: Quantum computers can search rapidly by using almost any transformation. Phys. Rev. Lett. 80(19), 4329-4332 (1998)
35. Hein, B., Tanner, G.: Quantum search algorithms on a regular lattice. Phys. Rev. A 82(1), 012326 (2010)
36. Hoffman, K.M., Kunze, R.: Linear Algebra. Prentice Hall, New York (1971)
37. Horn, R., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
38. Hughes, B.D.: Random Walks and Random Environments: Random Walks (Vol 1). Clarendon Press, Oxford (1995)
39. Hughes, B.D.: Random Walks and Random Environments: Random Environments (Vol 2). Oxford University Press, Oxford (1996)
40. Itakura, Y.K.: Quantum algorithm for commutativity testing of a matrix set. Master's thesis, University of Waterloo, Waterloo (2005)
41. Kaye, P., Laflamme, R., Mosca, M.: An Introduction to Quantum Computing. Oxford University Press, Oxford (2007)
42. Kempe, J.: Quantum random walks -an introductory overview. Contemp. Phys. 44(4), 302-327 (2003) quant-ph/0303081
43. Kempe, J.: Discrete quantum walks hit exponentially faster. Probab. Theor. Relat. Field. 133(2), 215-235 (2005), quant-ph/0205083
44. Konno, N.: Quantum random walks in one dimension. Quant. Inform. Process. 1(5), 345-354 (2002)
45. Košík, J.: Two models of quantum random walk. Cent. Eur. J. Phys. 4, 556-573 (2003)
46. Krovi, H., Magniez, F., Ozols, M., Roland, J.: Finding is as easy as detecting for quantum walks. In: Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 6198, pp. 540-551. Springer, Berlin (2010)
47. Lovász, L.: Random walks on graphs: a survey. Bolyai Society Mathematical Studies, Vol. 2, pp. 1-46. Springer (1993)
48. Lovett, N.B., Cooper, S., Everitt, M., Trevers, M., Kendon, V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81, 042330 (2010)
49. Mackay, T.D., Bartlett, S.D., Stephenson, L.T., Sanders, B.C.: Quantum walks in higher dimensions. J. Phys. A: Math. General 35(12), 2745 (2002)
50. Magniez, F., Nayak, A.: Quantum complexity of testing group commutativity. Algorithmica 48(3), 221-232 (2007)
51. Magniez, F., Nayak, A., Richter, P., Santha, M.: On the hitting times of quantum versus random walks. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 86-95. Philadelphia (2009)
52. Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, pp. 575-584. New York (2007)
53. Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. SIAM J. Comput. 37(2), 413-424. New York (2007)
54. Marquezino, F.L., Portugal, R., Abal, G.: Mixing times in quantum walks on two-dimensional grids. Phys. Rev. A 82(4), 042341 (2010)
55. Marquezino, F.L., Portugal, R., Abal, G., Donangelo, R.: Mixing times in quantum walks on the hypercube. Phys. Rev. A 77, 042312 (2008)
56. Marquezino, F.L., Portugal, R.: The QWalk simulator of quantum walks. Comput. Phys. Commun. 179(5), 359-369 (2008), arXiv:0803.3459
57. Mermin, N.D.: Quantum Computer Science: An Introduction. Cambridge University Press, New York (2007)
58. Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2001)
59. Moore, C., Russell, A.: Quantum walks on the hypercube. In: Rolim, J.D.P., Vadhan, S. (eds.) Proceedings of Random 2002, pp. 164-178. Springer, Cambridge (2002)
60. Moore, C., Mertens, S.: The Nature of Computation. Oxford University Press, New York (2011)
61. Mosca, M.: Counting by quantum eigenvalue estimation. Theor. Comput. Sci. 264(1), 139-153 (2001)
62. Motwani, R., Raghavan, P.: Randomized algorithms. ACM Comput. Surv. 28(1), 33-37 (1996)
63. Nayak, A., Vishwanath, A.: Quantum walk on a line. DIMACS Technical Report 2000-43, quant-ph/0010117 (2000)
64. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, New York (2000)
65. Omn`es, R.: Understanding Quantum Mechanics. Princeton University Press, Princeton (1999)
66. Peres, A.: Quantum Theory: Concepts and Methods. Springer, Berlin (1995)
67. Preskill, J.: Lecture Notes on Quantum Computation. http://www.theory.caltech.edu/preskill/ ph229 (1998)
68. Rieffel, E., Polak, W.: Quantum Computing, a Gentle Introduction. MIT, Cambridge (2011)
69. Sakurai, J.J.: Modern Quantum Mechanics. Addison Wesley, Reading (1993)
70. Santos, R.A.M., Portugal, R.: Quantum hitting time on the complete graph. Int. J. Quant. Inform. 8(5), 881-894 (2010), arXiv:0912.1217
71. Shenvi, N., Kempe, J.,Whaley, K.B.: A quantum random walk search algorithm. Phys. Rev. A 67(5), 052307 (2003), quant-ph/0210064
72. Strang, G.: Linear Algebra and Its Applications. Brooks Cole, Belmont (1988)
73. Strauch, F.W.: Connecting the discrete-and continuous-time quantum walks. Phys. Rev. A 74(3), 030301 (2006)
74. Szegedy, M.: Quantum speed-up of markov chain based algorithms. In: Proceedings of the Fourty-fifth Annual IEEE Symposium on the Foundations of Computer Science, pp. 32-41 (2004). DOI: 10.1109/FOCS.2004.53
75. Szegedy, M.: Spectra of Quantized Walks and √δε Rule. (2004), quant-ph/0401053
76. Travaglione, B.C.,Milburn, G.J.: Implementing the quantum random walk. Phys. Rev. A 65(3), 032310 (2002)
77. Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5(1), 83 (2003), quant-ph/0304204
78. Tulsi, A.: Faster quantum-walk algorithm for the two-dimensional spatial search. Phys. Rev. A 78(1), 012310 (2008)
79. Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Information Processing 11(5), 1015-1106 (2012), arXiv:1201.4780
80. Venegas-Andraca, S.E.: Quantum Walks for Computer Scientists. Morgan and Claypool Publishers, San Rafael (2008)
81. Stefăńak, M., Kolĺar, B., Kiss, T., Jex, I.: Full revivals in 2d quantum walks. Phys. Scripta 2010(T140), 014035 (2010)
82. Stefăńak, M.: Interference phenomena in quantum information. PhD thesis, Czech Technical University (2010), arXiv:1009.0200
83. Zalka, C.: Grover's Quantum Searching Algorithm is Optimal. Phys. Rev. A 60, 2746-2751 (1999)