Limitations of quantum coset states for graph isomorphism

Proceedings of the Annual ACM Symposium on Theory of Computing

Volumn 2006, Issue , 2006, Pages 604-617

  Hallgren, Sean ^a   Moore, Cristopher ^b   Rötteler, Martin ^a   Russell, Alexander ^c   Sen, Pranab ^a  

^a NEC LABORATORIES AMERICA   (United States)

^b UNIVERSITY OF NEW MEXICO   (United States)

^c UNIVERSITY OF CONNECTICUT   (United States)

Author keywords

Graph isomorphism; Hidden subgroup problem; Quantum algorithms

Indexed keywords

COMPUTATION THEORY; GRAPH THEORY; PROBLEM SOLVING; THEOREM PROVING;

GRAPH ISOMORPHISM; HIDDEN SUBGROUP PROBLEM (HSP); QUANTUM ALGORITHMS;

QUANTUM THEORY;

EID: 33748123212     PISSN: 07378017     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1132516.1132603     Document Type: Conference Paper

References (39)

1. D. Aharonov and A. Ta-Shma. Adiabatic quantum state generation and statistical zero knowledge. In Proceedings of the 35th Annual ACM Symposium on Theory of computing, pages 20-29, 2003. Also: ArXiv preprint quant-ph/0301023.
2. n. ArXiv preprint quant-ph/0511054, 2005.
3. D. Bacon, A. Childs, and W. v. Dam. From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005. Also: ArXiv preprint quant-ph/0504083.
4. R. Beals. Quantum computation of Fourier transforms over the symmetric groups. In Proceedings of the Symposium on Theory of Computing (STOC'97), pages 48-53, El Paso, Texas, 1997.
5. C. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computing. SIAM J. Comput., 26(5):1510-1523, 1997.
6. Y. G. Berkovich and E. M. Zhmud. Characters of finite groups, part 2, volume 181 of Translations of Mathematical Monographs. American Mathematical Society, 1999.
7. G. Brassard and P. Høyer. An exact polynomial-time algorithm for Simon's problem. In Proceedings of Fifth Israeli Symposium on Theory of Computing and Systems, pages 12-33. ISTCS, IEEE Computer Society Press, 1997. Also: ArXiv preprint quant-ph/9704027.
8. A. Childs and P. Wocjan. On the quantum hardness of solving isomorphism problems as nonabelian hidden shift problems. ArXiv preprint quant-ph/0510185, 2005.
9. M. Ettinger and P. Høyer. On quantum algorithms for noncommutative hidden subgroups. Advances in Applied Mathematics, 25(3): 239-251, 2000.
10. M. Ettinger, P. Høyer, and E. Knill. A quantum observable for the graph isomorphism problem. ArXiv preprint quant-ph/9901029, 1999.
11. M. Ettinger, P. Høyer, and E. Knill. Hidden subgroup states are almost orthogonal. ArXiv preprint quant-ph/9901034, 1999.
12. K. Friedl, G. Ivanyos, F. Magniez, M. Santha, and P. Sen. Hidden translation and orbit coset in quantum computing. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pages 1-9, 2003. Also: ArXiv preprint quant-ph/0211091.
13. W. Fulton and J. Harris. Representation theory: A first course, volume 129 of Graduate Texts in Mathematics. Springer, 1991.
14. P. X. Gallagher. Character values at involutions. Proceeedings of the American Mathematical Society, 120(3): 657-659, 1994.
15. M. Grigni, L. Schulman, M. Vazirani, and U. Vazirani. Quantum mechanical algorithms for the nonabelian hidden subgroup problem. Combinatorial, pages 137-154, 2004.
16. L. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 212-219, 1996. Also: ArXiv preprint quant-ph/9605043.
17. S. Hallgren. Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem. In Proceedings of the 34th Annual ACM Symposium on Theory of computing, pages 653-658, 2002.
18. S. Hallgren. Fast quantum algorithms for computing the unit group and class group of a number field. In Proceedings of the 87th Annual ACM Symposium on Theory of Computing, pages 468-474, 2005.
19. S. Hallgren, M. Rötteler, and P. Sen. Limitations of quantum coset states for graph isomorphism. arXiv preprint quant-ph/0511148.
20. S. Hallgren, A. Russell, and A. Ta-Shma. The hidden subgroup problem and quantum computation using group representations. SIAM Journal on Computing, 32(4):916-934, 2003.
21. I. M. Isaacs. Character theory of finite groups. Academic Press, 1976.
22. G. Ivanyos, F. Magniez, and M. Santha. Efficient quantum algorithms for some instances of the non-abelian hidden subgroup problem. International Journal of Foundations of Computer Science, pages 723-740, 2003. Also: ArXiv preprint quant-ph/0102014.
23. G. James and A. Kerber. The representation theory of the symmetric group. Addison-Wesley, Reading, 1981.
24. A. Y. Kitaev. Quantum measurements and the abelian stabilizer problem. ArXiv preprint quant-ph/9511026, 1995.
25. J. Köbler, U. Scheming, and J. Torán. The graph isomorphism problem. Birkhäuser, 1993.
26. G. Kuperberg. A subexponential-time quantum algorithm for the dihedral hidden subgroup problem. ArXiv preprint quant-ph/0302112, 2003.
27. J. D. Lafferty and D. Rockmore. Fast Fourier analysis for SL2 over a finite field and related numerical experiments. Experimental Mathematics, 1(2): 115-139, 1992.
28. C. Moore, D. Rockmore, A. Russell, and L. Schulman. The power of basis selection in fourier sampling: Hidden subgroup problems in affine groups. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1113-1122, 2004. Journal version in preparation. Also: ArXiv preprint quant-ph/0503095.
29. C. Moore and A. Russell. The symmetric group defies strong Fourier sampling: Part II. ArXiv preprint quant-ph/0501066, 2005.
30. C. Moore, A. Russell, and L. Schulman. The symmetric group defies strong Fourier sampling. In Proceedings of the 46th Annual IEEE Symposium on the Foundations of Computer Science, pages 479-488, 2005. Also: ArXiv preprint quant-ph/0501056.
31. M. Mosca and A. Ekert. The hidden subgroup problem and eigenvalue estimation on a quantum computer. In Quantum Computing and Quantum Communications, volume 1509 of Lecture Notes in Computer Science, pages 174-188. Springer-Verlag, 1998.
32. M. Nielsen and I. Chuang. Quantum computation and quantum information. Cambridge University Press, 2000.
33. J. Radhakrishnan, M. Rötteler, and P. Sen. On the power of random bases in Fourier sampling: Hidden subgroup problem in the Heisenberg group. In Proceedings of the Send International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 3580, pages 1399-1411. Springer-Verlag, 2005. Also: ArXiv preprint quant-ph/0503114.
34. Y. Roichman. Upper bound on the characters of the symmetric groups. Inventiones Mathematicae, 125:451-485, 1996.
35. A. Schmidt and U. Vollmer. Polynomial time quantum algorithm for the computation of the unit group of a number field. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 475-480, 2005.
36. P. Sen. Random measurement bases, quantum state distinction and applications to the hidden subgroup problem. In Proceedings of the 21st IEEE Conference on Computational Complexity, 2006. Tb appear. Also: ArXiv preprint quant-ph/0512085.
37. J. P. Serre. Linear representations of finite groups. Springer, 1977.
38. P. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5).1484-1509, 1997.
39. D. R. Simon. On the power of quantum computation. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pages 116-123, Los Alamitos, CA, 1994. Institute of Electrical and Electronic Engineers Computer Society Press.