Skip to main content
Springer Nature Link
Log in

Unbounded-Error Quantum Query Complexity

  • Conference paper
Algorithms and Computation (ISAAC 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5369))

Included in the following conference series:

  • 1630 Accesses

Abstract

This work studies the quantum query complexity of Boolean functions in an unbounded-error scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We first show that, just as in the communication complexity model, the unbounded-error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Next, connecting the query and communication complexity results, we show that the “black-box” approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve-Wigderson [STOC’98] is optimal even in the unbounded-error setting. We also study a related setting, called the weakly unbounded-error setting. In contrast to the case of communication complexity, we show a tight multiplicative Θ(logn) separation between quantum and classical query complexity in this setting for a partial Boolean function.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Aaronson, S., Ambainis, A.: Quantum search of spatial regions. Theory of Computing 1, 47–79 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambainis, A.: A note on quantum black-box complexity of almost all Boolean functions. Inform. Process. Lett. 71, 5–7 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon, N., Spencer, J.: The probabilistic method. Discrete Mathematics and Optimization. Wiley Interscience, Hoboken (2000)

    Book  MATH  Google Scholar 

  4. Aspnes, J., Beigel, R., Furst, M., Rudich, S.: The expressive power of voting polynomials. Combinatorica 14, 1–14 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity. In: Proc. 27th FOCS, pp. 303–312 (1986)

    Google Scholar 

  6. Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48, 778–797 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beigel, R.: Perceptrons, PP, and the polynomial hierarchy. Comput. Complexity 4, 339–349 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26, 1411–1473 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buhrman, H., Cleve, R., Wigderson, A.: Quantum vs. classical communication and computation. In: Proc. 30th STOC, pp. 63–68 (1998)

    Google Scholar 

  10. Buhrman, H., Vereshchagin, N., de Wolf, R.: On computation and communication with small bias. In: Proc. 22nd CCC, pp. 24–32 (2007)

    Google Scholar 

  11. van Dam, W.: Quantum oracle interrogation: getting all information for almost half the price. In: Proc. 39th FOCS, pp. 362–367 (1998)

    Google Scholar 

  12. O’Donnell, R., Servedio, R.A.: Extremal properties of polynomial threshold functions. In: Proc. 18th CCC, pp. 3–12 (2003)

    Google Scholar 

  13. Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: A limit on the speed of quantum computation in determining parity. Phys. Rev. Lett. 81, 5442–5444 (1998)

    Article  Google Scholar 

  14. Forster, J.: A linear lower bound on the unbounded error probabilistic communication complexity. J. Comput. Syst. Sci. 65, 612–625 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Le Gall, F.: Quantum weakly nondeterministic communication complexity. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 658–669. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  16. Halstenberg, B., Reischuk, R.: Relations between communication complexity classes. J. Comput. Syst. Sci. 41, 402–429 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Iwama, K., Nishimura, H., Raymond, R., Yamashita, S.: Unbounded-error one-way classical and quantum communication complexity. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 110–121. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  18. Iwama, K., Nishimura, H., Raymond, R., Yamashita, S.: Unbounded-error classical and quantum communication complexity. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 100–111. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  19. Klauck, H.: Lower bounds for quantum communication complexity. SIAM J. Comput. 37, 20–46 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Massar, S., Bacon, D., Cerf, N., Cleve, R.: Classical simulation of quantum entanglement without local hidden variables. Phys. Rev. A 63, 052305 (2001)

    Article  Google Scholar 

  21. Minsky, M.L., Papert, S.A.: Perceptrons. MIT Press, Cambridge (1988)

    MATH  Google Scholar 

  22. Paturi, R., Simon, J.: Probabilistic communication complexity. J. Comput. Syst. Sci. 33, 106–123 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Razborov, A.A.: Quantum communication complexity of symmetric predicates. Izvestiya Math. (English version) 67, 145–149 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sherstov, A.: Halfspace matrices. In: Proc. 22nd CCC, pp. 83–95 (2007)

    Google Scholar 

  25. de Wolf, R.: Quantum Computing and Communication Complexity, University of Amsterdam (2001)

    Google Scholar 

  26. de Wolf, R.: Nondeterministic quantum query and communication complexities. SIAM J. Comput. 32, 681–699 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Bristol, UK

    Ashley Montanaro

  2. School of Science, Osaka Prefecture University, Japan

    Harumichi Nishimura

  3. Tokyo Research Laboratory, IBM Japan, Japan

    Rudy Raymond

Authors
  1. Ashley Montanaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Harumichi Nishimura
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rudy Raymond
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Information Technologies, University of Sydney, Australia

    Seok-Hee Hong

  2. Graduate School of Informatics, Kyoto University, 606-8501, Kyoto, Japan

    Hiroshi Nagamochi

  3. Graduate School of Informatics, Kyoto University, Japan

    Takuro Fukunaga

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Montanaro, A., Nishimura, H., Raymond, R. (2008). Unbounded-Error Quantum Query Complexity. In: Hong, SH., Nagamochi, H., Fukunaga, T. (eds) Algorithms and Computation. ISAAC 2008. Lecture Notes in Computer Science, vol 5369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92182-0_80

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-92182-0_80

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-92181-3

  • Online ISBN: 978-3-540-92182-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics