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Quantum Information Processing

, Volume 4, Issue 5, pp 355–386 | Cite as

Improved Bounds on Quantum Learning Algorithms

  • Alp Atici
  • Rocco A. Servedio
Article

Abstract

In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples.

  • Hunziker et al.[Quantum Information Processing, to appear] conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is \(O(\frac{\log |C|}{\sqrt{{\hat{\gamma}}^{C}}})\), where \(\hat{\gamma}^{C}\) is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using \(O(\frac{\log |C| \log \log |C|}{\sqrt{{\hat{\gamma}}^{C}}})\) quantum black-box queries.

  • We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other.

  • Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for ɛ, Δ-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain \(\{0,1\}^n\) from \(\Omega({\frac d n})\) to \(\Omega(\frac{1}{\epsilon}\log \frac{1}{\delta}+d+\frac{\sqrt{d}}{\epsilon})\). This new lower bound comes closer to matching known upper bounds for classical PAC learning.

Keywords

Quantum query algorithms quantum computation computational learning theory PAC learning 

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References

  1. 1.
    Bennett C., Bernstein E., Brassard G., Vazirani U. (1997). SIAM J. Comput 26(5m):1510zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Grover L.K. (1996). Proceedings of the 28th ACM Symposium on Theory of Computing., pp. 212–219Google Scholar
  3. 3.
    Deutsch D., Josza R. (1992). Proc. Royal Soc. London A 439:553zbMATHADSCrossRefGoogle Scholar
  4. 4.
    Simon D.R. (1997). SIAM J. Comput 26(5):1474zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Servedio R.A., Gortler S.J. (2004). SIAM J. Comput 33(5):1067zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Angluin D. (1988). Machine Learning 2:319Google Scholar
  7. 7.
    Hunziker M., Meyer D.A., J. Park, J. Pommersheim, and M. Rothstein, arXiv: quant-ph/0309059 to appear in Quantum Information Processing Google Scholar
  8. 8.
    Ambainis A., Iwama K., Kawachi A., Masuda H., Putra R.H., Yamashita S. (2004). Proceedings of STACS pp. 93–104Google Scholar
  9. 9.
    Bshouty N.H., Jackson J.C. (1999). SIAM J. Comput 28(3):1136zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Valiant L.G. (1984). Comm. ACM 27:1134zbMATHCrossRefGoogle Scholar
  11. 11.
    Ehrenfeucht A., Haussler D., Kearns M., Valiant L. (1989). Informat Comput 82:247zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Blumer A., Ehrenfeucht A., Haussler D., Warmuth M. (1989). J. Assoc. Comput. Mach 36(4):929zbMATHMathSciNetGoogle Scholar
  13. 13.
    Bshouty N., Cleve R., Gavaldà R., Kannan S., Tamon C. (1996). J. Comput. Syst. Sci 52(3):421zbMATHCrossRefGoogle Scholar
  14. 14.
    R. Gavaldà. Proc. Ninth Structure in Complexity Theory Conference., 324 (1994)Google Scholar
  15. 15.
    Heged˝s T. (1995). Proc. Eighth Conf. on Computational Learning Theory pp. 108–117Google Scholar
  16. 16.
    Hellerstein L., Pillaipakkamnatt K., Raghavan V., Wilkins D. (1996). J. ACM 43(5):840–862zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Beals R., Buhrman H., Cleve R., Mosca M., de Wolf R. Proceedings of the 39th IEEE Symposium on Foundation of Computer Science., pp. 352–361 (1998)Google Scholar
  18. 18.
    Farhi E., Goldstone J., Gutmann S., Sipser M. (1998). Phys. Rev. Lett 81:5442CrossRefADSGoogle Scholar
  19. 19.
    Boyer M., Brassard G., Høyer P., Tapp A. (1998). Fortschritte der Physik 46(4–5):493CrossRefADSGoogle Scholar
  20. 20.
    Iwama K., Kawachi A., Raymond R., Yamashita S. arXiv:quant-ph/0411204 (2005)Google Scholar
  21. 21.
    Bernstein E., Vazirani U. (1997). SIAM J. Comput 26(5):1411zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    van Dam W. Proceedings of the 39th IEEE Symposium on Foundation of Computer Science., pp. 362–367 (1998)Google Scholar
  23. 23.
    Kearns M., Vazirani U. An Introduction to Computational Learning Theory. MIT Press, 1994Google Scholar
  24. 24.
    Shi Y. (2000). Informat. Processing Lett 75:79CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media 2005

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of Computer ScienceColumbia UniversityNew YorkUSA

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