Search via quantum walk

SIAM Journal on Computing

Volumn 40, Issue 1, 2011, Pages 142-164

  Magniez, Fréd́ric ^a   Nayak, Ashwin ^b   Roland, J́ŕmie ^c   Santha, Miklos ^c  

^a UNIVERSITÉ PARIS DIDEROT   (France)

^b UNIVERSITY OF WATERLOO   (Canada)

^c NEC LABORATORIES AMERICA   (United States)

Author keywords

Amplitude amplification; Hitting time; Markov chain; Phase estimation; Quantum walk; Search

Indexed keywords

AMPLITUDE AMPLIFICATION; HITTING TIME; MARKOV CHAIN; PHASE ESTIMATION; QUANTUM WALK; SEARCH;

ALGORITHMS; AMPLIFICATION; COMPUTER SCIENCE; ESTIMATION; MARKOV PROCESSES; MATHEMATICAL OPERATORS; PRESSES (MACHINE TOOLS);

QUANTUM THEORY;

EID: 79952954183     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/090745854     Document Type: Article

References (32)

1. S. Aaronson and Y. Shi, Quantum lower bounds for the collision and the element distinctness problems, J. ACM, (2004), pp. 595-605.
2. D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani, Quantum walks on graphs, in Proceedings of the 33rd ACM Symposium on Theory of Computing, 2001, pp. 50-59. (Pubitemid 32865717)
3. D. Aldous and J. A. Fill, Reversible Markov Chains and Random Walks on Graphs, monograph in preparation, 2006; available online at http://www.stat. berkeley.edu/users/ aldous/RWG/book.html.
4. A. Ambainis, Quantum walk algorithm for Element Distinctness, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 2004, pp. 22-31. (Pubitemid 40575263)
5. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, One-dimensional quantum walks, in Proceedings of the 33rd ACM Symposium on Theory of Computing, 2001, pp. 37-49. (Pubitemid 32865716)
6. A. Ambainis, J. Kempe, and A. Rivosh, Coins make quantum walks faster, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, 2005, pp. 1099-1108. (Pubitemid 40851478)
7. C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, Strengths and weaknesses of quantum computing, SIAM J. Comput., 26 (1997), pp. 1510-1523. (Pubitemid 127613101)
8. R. Bhatia, Matrix Analysis, Grad. Texts in Math. 169, Springer, New York, 1993.
9. M. Boyer, G. Brassard, P. Høyer, and A. Tapp, Tight bounds on quantum searching, Fortschr. Phys., 46 (1998), pp. 493-505. (Pubitemid 128186540)
10. G. Brassard, P. Høyer, M. Mosca, and A. Tapp, Quantum amplitude amplification and estimation, in Quantum Computation and Quantum Information: A Millennium Volume, J. S. J. Lomonaco and H. E. Brandt, eds., Contemp. Math. 305, AMS, Providence, RI, 2002, pp. 53-74.
11. A. Z. Broder and A. R. Karlin, Bounds on the cover time, J. Theoret. Probab., 2 (1989), pp. 101-120.
12. H. Buhrman, C. Dürr, M. Heiligman, P. Høyer, F. Magniez, M. Santha, and R. de Wolf, Quantum algorithms for element distinctness, SIAM J. Comput., 34 (2005), pp. 1324-1330. (Pubitemid 43003143)
13. H. Buhrman and R. Špalek, Quantum verification of matrix products, in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, 2006, pp. 880-889. (Pubitemid 43275312)
14. R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Quantum algorithms revisited, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), pp. 339-354. (Pubitemid 128733166)
15. S. Dörn and T. Thierauf, The quantum query complexity of algebraic properties, in Proceedings of the 16th International Symposium on the Fundamentals of Computation Theory, Lecture Notes in Comput. Sci. 4639, Springer, Berlin, Heidelberg, 2007, pp. 250-260.
16. L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the 28th ACM Symposium on Theory of Computing, 1996, pp. 212-219.
17. P. Høyer, M. Mosca, and R. de Wolf, Quantum search on bounded-error inputs, in Proceedings of the 30th International Colloquium on Automata, Languages and Programming, Lecture Notes in Comput. Sci. 2719, Springer, Berlin, Heidelberg, 2003, pp. 291-299.
18. C. Jordan, Essai sur la géométrie à n dimensions, Bull. Soc. Math. France, 3 (1875), pp. 103-174.
19. A. Kitaev, Quantum Measurements and the Abelian Stabilizer Problem, Technical Report, 1995; available online at http://arxiv.org/abs/quant-ph/ 9511026.
20. A. Y. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and Quantum Computation, Grad. Stud. Math. 47, AMS, Providence, RI, 2002.
21. H. Krovi, F. Magniez, M. Ozols, and J. Roland, Finding is as easy as detecting for quantum walks, in Proceedings of 37th International Colloquium on Automata, Languages and Programming, Lecture Notes in Comput. Sci. 6198, Springer, Berlin, Heidelberg, 2010, pp. 540-551.
22. F. Magniez and A. Nayak, Quantum complexity of testing group commutativity, Algorithmica, 48 (2007), pp. 221-232. (Pubitemid 47145791)
23. F. Magniez, A. Nayak, P. C. Richter, and M. Santha, On the hitting times of quantum versus random walks, in Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, 2009, pp. 86-95.
24. F. Magniez, M. Santha, and M. Szegedy, Quantum algorithms for the triangle problem, SIAM J. Comput., 37 (2007), pp. 413-424. (Pubitemid 351584851)
25. D. Meyer, From quantum cellular automata to quantum lattice gases, J. Statist. Phys., 85 (1996), pp. 551-574.
26. A. Nayak and A. Vishwanath, Quantum Walk on the Line, Technical Report, 2000; available online at http://arxiv.org/abs/quant-ph/0010117.
27. P. C. Richter, Almost uniform sampling via quantum walks, New J. Phys., 9 (2007), article 72.
28. N. Shenvi, J. Kempe, and K. B. Whaley, Quantum random-walk search algorithm, Phys. Rev. A, 67 (2003), article 052307.
29. M. Szegedy, Quantum speed-up of Markov chain based algorithms, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 2004, pp. 32-41. (Pubitemid 40575264)
30. A. Tulsi, Faster quantum walk algorithm for the two dimensional spatial search, Phys. Rev. A, 78 (2008), article 012310.
31. U. Vazirani, On the power of quantum computation, Philos. Trans. Roy. Soc. London Ser. A, 356 (1998), pp. 1759-1768. (Pubitemid 128542383)
32. J.Watrous, Quantum simulations of classical random walks and undirected graph connectivity, J. Comput. System Sci., 62 (2001), pp. 376-391. (Pubitemid 32334557)