Advertisement

Weak Parity

  • Scott Aaronson
  • Andris Ambainis
  • Kaspars Balodis
  • Mohammad Bavarian
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

Abstract

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2 + ε fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity on all inputs. But surprisingly, we give a randomized algorithm for Weak Parity that makes only O(n/log0.246(1/ε)) queries, as well as a quantum algorithm that makes \(O(n/\sqrt{\log(1/\varepsilon)})\) queries. We also prove a lower bound of \(\Omega\left( n/\log\left( 1/\varepsilon\right) \right) \) in both cases, as well as lower bounds of Ω(logn) in the randomized case and \(\Omega(\sqrt{\log n})\) in the quantum case for any ε > 0. We show that improving our lower bounds is intimately related to two longstanding open problems about Boolean functions: the Sensitivity Conjecture, and the relationships between query complexity and polynomial degree.

Keywords

Boolean Function Quantum Algorithm Query Complexity Full Version Quantum Walk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambainis, A., Childs, A., Le Gall, F., Tani, S.: The quantum query complexity of certification. Quantum Information and Computation 10(3-4) arXiv:0903.1291 (2010)Google Scholar
  2. 2.
    Ambainis, A., Childs, A.M., Reichardt, B.W., Špalek, R., Zhang, S.: Any AND-OR formula of size N can be evaluated in time N 1/2 + o(1) on a quantum computer. In: Proc. IEEE FOCS (2007); quant-ph/0703015 and arXiv:0704.3628Google Scholar
  3. 3.
    Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48(4), 778–797 (2001); Earlier version in IEEE FOCS 1998, pp. 352–361 (1998) quant-ph/9802049Google Scholar
  4. 4.
    Bennett, C., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM J. Comput. 26(5), 1510–1523 (1997); quant-ph/9701001Google Scholar
  5. 5.
    Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theoretical Comput. Sci. 288, 21–43 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chung, F.R.K., Füredi, Z., Graham, R.L., Seymour, P.: On induced subgraphs of the cube. J. Comb. Theory Ser. A 49(1), 180–187 (1988)CrossRefzbMATHGoogle Scholar
  7. 7.
    Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. Roy. Soc. London A439, 553–558 (1992)Google Scholar
  8. 8.
    Farhi, E., Goldstone, J., Gutmann, S.: A quantum algorithm for the Hamiltonian NAND tree. Theory of Computing 4(1), 169–190 (2008); quant-ph/0702144Google Scholar
  9. 9.
    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); quant-ph/9802045Google Scholar
  10. 10.
    Gotsman, C., Linial, N.: The equivalence of two problems on the cube. J. Comb. Theory Ser. A 61(1), 142–146 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hatami, P., Kulkarni, R., Pankratov, D.: Variations on the sensitivity conjecture. Theory of Computing Library Graduate Surveys 4 (2011)Google Scholar
  12. 12.
    Midrijanis, G.: On randomized and quantum query complexities (2005) quant-ph/0501142Google Scholar
  13. 13.
    Nisan, N.: CREW PRAMs and decision trees. SIAM J. Comput. 20(6), 999–1007 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Nisan, N., Szegedy, M.: On the degree of Boolean functions as real polynomials. Computational Complexity 4(4), 301–313 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Saks, M., Wigderson, A.: Probabilistic Boolean decision trees and the complexity of evaluating game trees. In: Proc. IEEE FOCS, pp. 29–38 (1986)Google Scholar
  16. 16.
    Santha, M.: On the Monte-Carlo decision tree complexity of read-once formulae. Random Structures and Algorithms 6(1), 75–87 (1995)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Tal, A.: Properties and applications of Boolean function composition. In: Proc. Innovations in Theoretical Computer Science (ITCS), pp. 441–454 (2013); ECCC TR12-163Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Scott Aaronson
    • 1
  • Andris Ambainis
    • 2
  • Kaspars Balodis
    • 2
  • Mohammad Bavarian
    • 1
  1. 1.MITUSA
  2. 2.University of LatviaLatvia

Personalised recommendations