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On the Complexity of the Hidden Subgroup Problem

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Theory and Applications of Models of Computation (TAMC 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4978))

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Abstract

We show that several problems that figure prominently in quantum computing, including Hidden Coset, Hidden Shift, and Orbit Coset, are equivalent or reducible to Hidden Subgroup. We also show that, over permutation groups, the decision version and search version of Hidden Subgroup are polynomial-time equivalent. For Hidden Subgroup over dihedral groups, such an equivalence can be obtained if the order of the group is smooth. Finally, we give nonadaptive program checkers for Hidden Subgroup and its decision version.

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References

  1. Arvind, V., Kurur, P.P.: Graph Isomorphism is in SPP. In: Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, IEEE, New York (2002)

    Google Scholar 

  2. Arvind, V., Torán, J.: A nonadaptive NC checker for permutation group intersection. Theoretical Computer Science 259, 597–611 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Blum, M., Kannan, S.: Designing programs that check their work. Journal of the ACM 42(1), 269–291 (1995)

    Article  MATH  Google Scholar 

  4. Childs, A.M., Schulman, L.J., Vazirani, U.V.: Quantum algorithms for hidden nonlinear structures. In: FOCS 2007: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), pp. 395–404. IEEE Computer Society, Washington (2007)

    Google Scholar 

  5. Childs, A.M., van Dam, W.: Quantum algorithm for a generalized hidden shift problem. In: SODA 2007: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 1225–1232 (2007)

    Google Scholar 

  6. Ettinger, M., Høyer, P.: On quantum algorithms for noncommutative hidden subgroups. Advances in Applied Mathematics 25, 239–251 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ettinger, M., Høyer, P., Knill, E.: The quantum query complexity of the hidden subgroup problem is polynomial. Information Processing Letters 91(1), 43–48 (2004)

    Article  MathSciNet  Google Scholar 

  8. Furst, M.L., Hopcroft, J.E., Luks, E.M.: Polynomial-time algorithms for permutation groups. In: Proceedings of the 21st IEEE Symposium on Foundations of Computer Science, pp. 36–41 (1980)

    Google Scholar 

  9. 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 the Theory of Computing, pp. 1–9 (2003)

    Google Scholar 

  10. Grigni, M., Schulman, J., Vazirani, M., Vazirani, U.: Quantum mechanical algorithms for the nonabelian hidden subgroup problem. Combinatorica 24(1), 137–154 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hallgren, S., Moore, C., Rötteler, M., Russell, A., Sen, P.: Limitations of quantum coset states for graph isomorphism. In: STOC 2006: Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pp. 604–617 (2006)

    Google Scholar 

  12. Hallgren, S., Russell, A., Ta-Shma, A.: The hidden subgroup problem and quantum computation using group representations. SIAM Journal on Computing 32(4), 916–934 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jozsa, R.: Quantum factoring, discrete algorithm and the hidden subgroup problem (manuscript, 2000)

    Google Scholar 

  14. Kitaev, A.Y.: Quantum measurements and the Abelian Stabilizer problem, quant-ph/9511026 (1995)

    Google Scholar 

  15. Kempe, J., Shalev, A.: The hidden subgroup problem and permutation group theory. In: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, Vancouver, British Columbia, January 2005, pp. 1118–1125 (2005)

    Google Scholar 

  16. Mathon, R.: A note on the graph isomorphism counting problem. Information Processing Letters 8, 131–132 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mosca, M.: Quantum Computer Algorithms. PhD thesis, University of Oxford (1999)

    Google Scholar 

  18. Moore, C., Russell, A., Schulman, L.J.: The symmetric group defies strong fourier sampling. In: FOCS 2005: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 479–490 (2005)

    Google Scholar 

  19. Moore, C., Russell, A., Sniady, P.: On the impossibility of a quantum sieve algorithm for graph isomorphism. In: STOC 2007: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pp. 536–545 (2007)

    Google Scholar 

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

    MATH  Google Scholar 

  21. Quantum pontiff, http://dabacon.org/pontiff/?p=899

  22. Regev, O.: Quantum computation and lattice problems. SIAM Journal on Computing 33(3), 738–760 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Scott, W.R.: Group Theory. Dover Publications, Inc. (1987)

    Google Scholar 

  24. Sims, C.C.: Computational methods in the study of permutation groups. In: Computational problems in abstract algebra, pp. 169–183 (1970)

    Google Scholar 

  25. van Dam, W., Hallgren, S., Ip, L.: Quantum algorithms for some hidden shift problems. In: Proceedings of the 14th annual ACM-SIAM symposium on Discrete algorithms, pp. 489–498 (2003)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science & Engineering, University of South Carolina, Columbia, SC 29208, USA

    Stephen Fenner

  2. Department of Mathematical Sciences, Eastern Mennonite University, Harrisonburg, VA 22802, USA

    Yong Zhang

Authors
  1. Stephen Fenner
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  2. Yong Zhang
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Editor information

Manindra Agrawal Dingzhu Du Zhenhua Duan Angsheng Li

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© 2008 Springer-Verlag Berlin Heidelberg

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Fenner, S., Zhang, Y. (2008). On the Complexity of the Hidden Subgroup Problem. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_6

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  • DOI: https://doi.org/10.1007/978-3-540-79228-4_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-79227-7

  • Online ISBN: 978-3-540-79228-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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