Quantum algorithms for element distinctness

SIAM Journal on Computing

Volumn 34, Issue 6, 2005, Pages 1324-1330

  Buhrman, Harry ^a   Dürr, Christoph ^b   Heiligman, Mark ^c   Høyer, Peter ^d   Magniez, Frédéric ^e   Santha, Miklos ^e   De Wolf, Ronald ^a  

^a CWI   (Netherlands)

^b UNIVERSITÉ PARIS SACLAY   (France)

^c NSA   (United States)

^d UNIVERSITY OF CALGARY   (Canada)

^e CNRS   (France)

Author keywords

Collision problem; Element distinctness; Quantum query complexity

Indexed keywords

COLLISION PROBLEM; ELEMENT DISTINCTNESS; QUANTUM ALGORITHMS; QUANTUM QUERY COMPLEXITY;

ALGORITHMS; GRAPH THEORY; MATHEMATICAL TECHNIQUES; SORTING;

QUANTUM THEORY;

EID: 29344468473     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539702402780     Document Type: Article

References (25)

1. S. AARONSON AND Y. SHI, Quantum lower bounds for the collision and the element distinctness problems, J. ACM, 51 (2004), pp. 595-605.
2. M. AJTAI, Determinism versus nondeterminism for linear time RAMs with memory restrictions, J. Comput. System Sci., 65 (2002), pp. 2-37.
3. A. AMBAINIS, Quantum Walk Algorithm for Element Distinctness, http://arxiv.org/abs/quant-ph/0311001 (1 Nov 2003).
4. A. AMBAINIS, Quantum Lower Bounds for Collision and Element Distinctness with Small Range, http://arxiv.org/abs/quant-ph/0305179 (29 May 2003).
5. P. BEAME, M. SAKS, X. SUN, AND E. VEE, Super-linear time-space tradeoff lower bounds for randomized computation, in Proceedings of the 41st IEEE Symposium on Foundations of Computer Science (FOCS), 2000, pp. 169-179.
6. G. BRASSARD AND P. HØYER, An exact quantum polynomial-time algorithm for Simon's problem, in Proceedings of the 5th Israeli Symposium on Theory of Computing and Systems (ISTCS), 1997, pp. 12-23.
7. G. BRASSARD, P. HØYER, M. MOSCA, AND A. TAPP, Quantum amplitude amplification and estimation, in Quantum Computation and Quantum Information: A Millennium Volume, Contemp. Math. 305, AMS, Providence, RI, 2002, pp. 53-74.
8. G. BRASSARD, P. HØYER, AND A. TAPP, Quantum algorithm for the collision problem, ACM SIGACT News, 28 (1997), pp. 14-19.
9. H. BUHRMAN, R. CLEVE, AND A. WIGDERSON, Quantum vs. classical communication and computation, in Proceedings of the 30th ACM Symposium on Theory of Computing (STOC), 1998. pp. 63-68.
10. H. BUHRMAN, R. CLEVE, R. DE WOLF, AND CH. ZALKA, Bounds for small-error and zero-error quantum algorithms, in Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), 1999, pp. 358-368.
11. H. BUHRMAN, C. DÜRR, M. HEILIGMAN, P. HØYER, F. MAGNIEZ, M. SANTHA, AND R. DE WOLF, Quantum algorithms for element distinctness, in Proceedings of the 16th IEEE Conference on Computational Complexity, 2001, pp. 131-137.
12. C. DÜRR, M. HEILIGMAN, P. HØYER, AND M. MHALLA, Quantum query complexity of some graph problems, in Proceedings of the 31st ICALP, Lecture Notes in Comput. Sci. 3142, Springer-Verlag, New York, 2004, pp. 481-493.
13. E. FARHI, J. GOLDSTONE, S. GUTMANN, AND M. SIPSER, Invariant Quantum Algorithms for Insertion into an Ordered List, http://arxiv.org/abs/quant-ph/ 9901059 (19 Jan 1999).
14. D. GRIGORIEV, Randomized complexity lower bounds, in Proceedings of the 30th ACM Symposium on Theory of Computing (STOC), 1998, pp. 219-223.
15. L. K. GROVER, A fast quantum mechanical algorithm for database search, in Proceedings of the 28th ACM Symposium on Theory of Computing (STOC), 1996, pp. 212-219.
16. P. HØYER, J. NEERBEK, AND Y. SHI, Quantum complexities of ordered searching, sorting, and element distinctness, in Proceedings of the 28th ICALP, Lecture Notes in Comput. Sci. 2076, Springer-Verlag, New York, 2001, pp. 346-357.
17. H. KLAUCK, R. SPALEK, AND R. DE WOLF, Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs, http://arxiv.org/abs/quant- ph/0402123 (18 Feb 2004).
18. A. LUBIW AND A. RÁCZ, A lower bound for the integer element distinctness problem, Inform. and Comput., 94 (1991), pp. 83-92.
19. 1.3) Quantum Algorithm for the Triangle Problem, http://arxiv.org/abs/quant-ph/0310134 (21 Oct 2003).
20. M. MOSCA, Quantum searching, counting, and amplitude amplification by eigenvector analysis, in MFCS Workshop on Randomized Algorithms, 1998, pp. 90-100.
21. A. NAYAK AND F. WU, The quantum query complexity of approximating the median and related statistics, in Proceedings of the 31st ACM Symposium on Theory of Computing (STOC), 1999. pp. 384-393.
22. M. A. NIELSEN AND I. L. CHUANG, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000.
23. P. W. SHOR, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput., 26 (1997), pp. 1484-1509.
24. A. C.-C. YAO, Near-optimal time-space tradeoff for element distinctness, SIAM J. Comput., 23 (1994), pp. 966-975.
25. A. C.-C. YAO, private communication, Department of Computer Science, Princeton University, Princeton, NJ, 2003.