Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-29T05:11:47.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Extending the Promise of the Deutsch–Jozsa–Høyer Algorithm for Finite Groups

Published online by Cambridge University Press:  01 February 2010

Michael Batty ,
Andrew J. Duncan  and
Samuel L. Braunstein
Michael Batty
Affiliation:
Department of Mathematics, School of Mathematics and Statistics, Merz Court, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, United Kingdom, [email protected], http://www.mas.ncl.ac.uk/~nmb45/
Andrew J. Duncan
Affiliation:
Department of Mathematics, School of Mathematics and Statistics, Merz Court, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, United Kingdom, [email protected], http://www.mas.ncl.ac.uk/~najd2/
Samuel L. Braunstein
Affiliation:
Department of Computer Science, University of York, York, YO10 5DD, United Kingdom. [email protected], http://www-users.cs.york.ac.uk/~schmuel/

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Høyer has given a generalisation of the Deutsch–Jozsa algorithm which uses the Fourier transform on a group G which is (in general) non-Abelian. His algorithm distinguishes between functions which are either perfectly balanced (m-to-one) or constant, with certainty, and using a single quantum query. Here, we show that this algorithm (which we call the Deutsch–Jozsa–Høyer algorithm) can in fact deal with a broader range of promises, which we define in terms of the irreducible representations of G.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 9 , 2006 , pp. 40 - 63
Copyright
Copyright © London Mathematical Society 2006

References

1. Beals, R., ‘Quantum computation of Fourier transforms over the symmetric groups’, Proc. 29th Annual ACM Symposium on Theory of Computing, El Paso, Texas, May 1997 (ACM, New York, 1999) 48–53.Google Scholar
2. Cleve, R., Ekert, A., Macchiavello, C. and Mosca, M., ‘Quantum algorithms revisited’, Proc. Royal Soc. London, Sen. A 454 (1998)339.Google Scholar
3. Constantini, G. and Smeraldi, F., ‘A generalisation of Deutsch's example’, 1997, http://lanl.arxiv.org/abs/quant-ph/9702020.Google Scholar
4. Coppersmith, D., ‘An Approximate Fourier transform useful for quantum factoring’, Technical Report RC 19642, IBM Research Division, Yorktown Heights, NY, December 1994.Google Scholar
5. Deutsch, D., ‘Quantum theory, ‘the Church–Turing principle and the universal quantum computer’, Proc. Royal Soc. London, Ser. A 400 (1985) 97–117.Google Scholar
6. Deutsch, D. and Jozsa, R., ‘Rapid solution of problems by quantum computation’, Proc. Royal Soc. London, Ser. A 439 (1992) 553–558.Google Scholar
7. Ettinger, M. and Høyer, P., ‘On quantum algorithms for noncommutative hidden subgroups’, Adv. Appl. Math. 25 (2000) 239–251.CrossRefGoogle Scholar
8. Fulton, W. and Harris, J., Representation theory, Grad. Texts in Math. 129 (Springer, New York, 1991).Google Scholar
9. Grigni, M., Schulman, L.J., Vazirani, M. and Vazirani, U., ‘Quantum mechanical algorithms for the nonabelian hidden subgroup problem’, STOC (2001), Combinatorica 24 (2004) 137–154.CrossRefGoogle Scholar
10. Hallgreen, S., Russell, A. and Ta-Shma, A., ‘Normal subgroup reconstruction and quantum computation using group representations’, Proc. 32nd Annual ACM Symposium on Theory of Computing (ACM, New York, 2000) 627–635.Google Scholar
11. Høyer, P., ‘Efficient quantum transforms’, 1997, http://lanl.arxiv.org/abs/quant-ph/9702028.Google Scholar
12. Høyer, P., ‘Conjugated operators in quantum algorithms’, Phys. Rev. A 59 (1999) 3280–3289.CrossRefGoogle Scholar
13. Ireland, K. and Rosen, M., A classical introduction to modern number theory, Grad. Texts in Math. 84 (Springer, New York, 1982).CrossRefGoogle Scholar
14. Ledermann, W., Introduction to group characters (Cambridge Univ. Press, Cambridge, 1987).CrossRefGoogle Scholar
15. Maslen, D.K. and Rockmore, D. N., ‘Generalised FFTs — a survey of some recent results’, Proc. 1995 DIMACS Workshop in Groups and- Computation, ed. Finkelstein, L. and Kantor, W., DIMACS Ser. Discrete Math. Comput. Sci. 28 (Amer. Math. Soc, Providence, RI, 1997) 183–238.Google Scholar
16. Moore, C., Rockmore, D. and Russell, A., ‘Generic quantum Fourier transforms’, Proc. Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), New Orleans, LA, January 11–13 2004 (SIAM, Philadelphia, PA, 2004) 778–787.Google Scholar
17. Nielsen, M.A. and Chuang, I.L., Quantum computation and quantum information (Cambridge Univ. Press, Cambridge UK, 2000).Google Scholar
18. Püschel, M., Rötteler, M. and Beth, T., ‘Fast quantum Fourier transforms for a class of non-abelian Groups’, Applied algebra, algebraic algorithms and error correcting codes, Proc. 13th international symposium, AAECC-13, Honolulu, HI, USA, November 15–19, 1999, ed. Fossorier, Marc et al. , Lect. Notes in Comput. Sci. 1719 (Springer, Berlin, 1999) 148–159.CrossRefGoogle Scholar
19. Rötteler, M. and Beth, T., ‘Polynomial-time solution to the hidden subgroup problem for a class of non-Abelian groups’, vl, 1998, http://lanl.arxiv.org/abs/quant-ph/9812070.Google Scholar
20. Shor, P.W., ‘Algorithms for quantum computation: discrete logarithm and factoring’, Proc. 35th Annual Symposium on Foundations of Computer Science (IEEE Computer Society Press, 1994) 124–134.CrossRefGoogle Scholar
21. Terras, A., Fourier analysis on finite groups and applications, London. Math. Soc. Student Texts 43 (Cambridge Univ. Press, Cambridge, UK, 1999).CrossRefGoogle Scholar