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An Efficient Quantum Algorithm for the Hidden Subgroup Problem in Nil-2 Groups

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Abstract

In this paper we show that the hidden subgroup problem in nil-2 groups, that is in groups of nilpotency class at most 2, can be solved efficiently by a quantum procedure. The algorithm is an extension of our earlier method for extraspecial groups in Ivanyos et al. (Proceedings of the 24th Symposium on Theoretical Aspects of Computer Science (STACS), vol. 4393, pp. 586–597, 2007), but it has several additional features. It contains a powerful classical reduction for the hidden subgroup problem in nilpotent groups of constant nilpotency class to the specific case where the group is a p-group of exponent p and the subgroup is either trivial or cyclic. This reduction might also be useful for dealing with groups of higher nilpotency class. The quantum part of the algorithm uses well chosen group actions based on some automorphisms of nil-2 groups. The right choice of the actions requires the solution of a system of quadratic and linear equations. The existence of a solution is guaranteed by the Chevalley-Warning theorem, and we prove that it can also be found efficiently.

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References

  1. Bacon, D., Childs, A., van Dam, W.: From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups. In: Proceedings of the 46th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 469–478 (2005)

    Chapter  Google Scholar 

  2. Beals, R., Babai, L.: Las Vegas algorithms for matrix groups. In: Proceedings of the 34th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 427–436 (1993)

    Google Scholar 

  3. Chevalley, C.: Démonstration d’une hypothèse de M. Artin. Abh. Math. Semin. Univ. Hamb. 11, 73–75 (1936)

    Article  Google Scholar 

  4. Eick, B.: Orbit-stabilizer problems and computing normalizers for polycyclic groups. J. Symb. Comput. 34, 1–19 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Friedl, K., Ivanyos, G., Magniez, F., Santha, M., Sen, P.: Hidden translation and orbit coset in quantum computing. In: Proceedings of the 35th ACM Symposium on Theory of Computing (STOC), pp. 1–9 (2003)

    Google Scholar 

  6. Gebhardt, V.: Efficient collection in infinite polycyclic groups. J. Symb. Comput. 34, 213–228 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Grigni, M., Schulman, L., Vazirani, M., Vazirani, U.: Quantum mechanical algorithms for the nonabelian Hidden Subgroup Problem. In: Proceedings of the 33rd ACM Symposium on Theory of Computing (STOC), pp. 68–74 (2001)

    Google Scholar 

  8. Hall, M., Jr.: Theory of Groups, 2nd edn. AMS Chelsea, Providence (1999)

    Google Scholar 

  9. Hallgren, S., Russell, A., Ta-Shma, A.: Normal subgroup reconstruction and quantum computation using group representations. SIAM J. Comput. 32(4), 916–934 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Holt, D.F., Eick, B., O’Brien, E.: Handbook of Computational Group Theory. Chapman & Hall/CRC Press, Boca Raton (2005)

    Book  MATH  Google Scholar 

  11. Höfling, B.: Efficient multiplication algorithms for finite polycyclic groups. Preprint, available at http://www-public.tu-bs.de/bhoeflin/preprints/collect.pdf (2004)

  12. Ivanyos, G., Magniez, F., Santha, M.: Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem. Int. J. Found. Comput. Sci. 14(5), 723–739 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ivanyos, G., Sanselme, L., Santha, M.: An efficient quantum algorithm for the hidden subgroup problem in extraspecial groups. In: Proceedings of the 24th Symposium on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Computer Science (LNCS), vol. 4393, pp. 586–597. Springer, Berlin (2007)

    Google Scholar 

  14. Kitaev, A.: Quantum measurements and the Abelian Stabilizer Problem. Technical report. arXiv:quant-ph/9511026 (1995)

  15. Krovi, H., Rötteler, M.: An efficient algorithm for the hidden subgroup problem in Weyl-Heisenberg groups. In: Proceedings of the 2008 Conference on Mathematical Methods in Computer Science (MMICS). Lecture Notes in Computer Science (LNCS), vol. 5393, pp. 70–88. Springer, Berlin (2008)

    Google Scholar 

  16. Leedham-Green, C.R., Soicher, L.H.: Collection from the left and other strategies. J. Symb. Comput. 9, 665–675 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  17. Leedham-Green, C.R., Soicher, L.H.: Symbolic collection using Deep Thought. LMS J. Comput. Math. 1, 9–24 (1998)

    MATH  MathSciNet  Google Scholar 

  18. Luks, E.M.: Computing in solvable matrix groups. In: Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 111–120 (1992)

    Chapter  Google Scholar 

  19. Mosca, M.: Quantum computer algorithms. PhD thesis, University of Oxford (1999)

  20. Meggido, N., Papadimitriou, C.: On total functions existence theorems, and computational complexity. Theor. Comput. Sci. 81, 317–324 (1991)

    Article  Google Scholar 

  21. Moore, C., Rockmore, D., Russell, A., Schulman, L.: The power of basis selection in Fourier sampling: Hidden subgroup problems in affine groups. In: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1106–1115 (2004)

    Google Scholar 

  22. Moore, C., Russell, A., Sniady, P.: On the impossibility of a quantum sieve algorithm for graph isomorphism. In: Proceedings of the 39th ACM Symposium on Theory of Computing (STOC), pp. 536–545 (2007)

    Google Scholar 

  23. Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  24. Robinson, D.J.S.: A Course in the Theory of Groups, 2nd edn. Springer, New York (1996)

    Book  Google Scholar 

  25. Rötteler, M., Beth, T.: Polynomial-time solution to the Hidden Subgroup Problem for a class of non-abelian groups. Technical report. arXiv:quant-ph/9812070 (1998)

  26. Regev, O.: Quantum computation and lattice problems. SIAM J. Comput. 33(3), 738–760 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Shanks, D.: Five number-theoretic algorithms. In: Proceedings of the 2nd Manitoba Conference on Numerical Mathematics, pp. 51–70 (1972)

    Google Scholar 

  28. Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  29. van de Woestijne, C.: Deterministic equation solving over finite fields. PhD thesis, Universiteit Leiden. Available at http://hdl.handle.net/1887/4392 (2006)

  30. Warning, E.: Bemerkung zur vorstehenden Arbeit von Herrn Chevalley. Abh Math. Semin. Univ. Hamb. 11, 76–83 (1936)

    Google Scholar 

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Authors and Affiliations

  1. Computer and Automation Research Institute of the Hungarian Academy of Sciences, Kende u. 13-17, 1111, Budapest, Hungary

    Gábor Ivanyos

  2. LRI, Université Paris–Sud, CNRS, 91405, Orsay, France

    Luc Sanselme & Miklos Santha

  3. Centre for Quantum Technologies, National University of Singapore, Singapore, 117543, Singapore

    Miklos Santha

Authors
  1. Gábor Ivanyos
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  2. Luc Sanselme
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  3. Miklos Santha
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Corresponding author

Correspondence to Gábor Ivanyos.

Additional information

Research supported by the European Commission Project Qubit Applications (QAP) 015848 and the FP7-ICT-2009-C Project Quantum Computer Science (QCS) 255961, the OTKA grants NK72845 and T77476, the French ANR QRAC project under contract ANR-08-EMER-012 and by the CNRS-MTA joint project 21434. Part of this work was conducted during the first author’s research visit at the Centre for Quantum Technologies, National University of Singapore. Research at the Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation. A preliminary version of this paper was presented at the 8th Latin American Theoretical Informatics Symposium (LATIN’08), Lecture Notes in Computer Science (LNCS), vol. 4957, pages 759–771, 2008.

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Ivanyos, G., Sanselme, L. & Santha, M. An Efficient Quantum Algorithm for the Hidden Subgroup Problem in Nil-2 Groups. Algorithmica 62, 480–498 (2012). https://doi.org/10.1007/s00453-010-9467-0

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