Skip to main content
Log in

Quantum Algorithms for Learning and Testing Juntas

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: (1) whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; (2) with no access to any classical or quantum membership (“black-box”) queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; (3) which require only a few quantum examples but possibly many classical random examples (which are considered quite “cheap” relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: (1) We give an algorithm for testing k-juntas to accuracy ε that uses O(k/ϵ) quantum examples. This improves on the number of examples used by the best known classical algorithm. (2) We establish the following lower bound: any FS-based k-junta testing algorithm requires \(\Omega(\sqrt{k})\) queries. (3) We give an algorithm for learning k-juntas to accuracy ϵ that uses O−1 k log k) quantum examples and O(2k log(1/ϵ)) random examples. We show that this learning algorithm is close to optimal by giving a related lower bound.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ambainis A., Iwama K., Kawachi A., Masuda H., Putra R.H., and Yamashita S., in Proceedings of STACS 2004, pp. 93–104 (2003).

  • Atıcı A., Servedio R.A. (2005). Quantum Information Process. 4, 355–386

    Google Scholar 

  • Bshouty N.H., Jackson J.C. (1999). SIAM J. Comp. 28, 1136–1153

    Article  MATH  MathSciNet  Google Scholar 

  • J. Castro, in Proceedings of 17th ALT, pp. 78–92 (2006).

  • M. Hunziker, Meyer D.A., J. Park, J. Pommersheim, and M. Rothstein, The Geometry of Quantum Learning, arXiv:quant-ph/0309059; Quantum Information Processing (to appear).

  • K. Iwama, A. Kawachi, R. Raymond, and S. Yamashita, Robust Quantum Algorithms for Oracle Identification, arXiv:quant-ph/0411204 (2005).

  • Servedio R.A., Gortler S.J. (2004). SIAM J. Comp. 33, 1067–1092

    Article  MATH  MathSciNet  Google Scholar 

  • H. Buhrman, L. Fortnow, I. Newman, and H. Röhrig, in Proceedings of 14th SODA, pp. 480–488 (2003).

  • K. Friedl, F. Magniez, M. Santha, and P. Sen, in Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science, pp. 419–428.

  • F. Magniez and A. Nayak, in Proceedings of the 32nd International Colloquium on Automata, Languages and Programming, pp. 1312–1324 (2005).

  • Jackson J.C. (1997). J. Comput. Syst. Sci. 55, 414–440

    Article  MATH  Google Scholar 

  • Valiant L.G. (1984). Commun. Assoc. Comp. Machinery 27, 1134–1142

    MATH  Google Scholar 

  • J. Arpe and R. Reischuk, in Proceedings of the 14th International Conference on Algorithmic Learning Theory, pp. 99–113 (2003).

  • J. Arpe and R. Reischuk, in Proceedings of the 3rd International Conference on Theory and Applications of Models of Computation, pp. 387–398 (2006).

  • A. Blum, in Proceedings of the 16th Annual Conference on Learning Theory and 7th Kernel Workshop, pp. 731–733 (2003).

  • Chockler H., Gutfreund D. (2004). Information Process. Lett. 90, 301–305

    Article  MathSciNet  Google Scholar 

  • E. Fischer, G. Kindler, D. Ron, S. Safra, and A. Samorodnitsky, in Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, pp. 103–112 (2002).

  • R. Lipton, E. Markakis, A. Mehta, and N. Vishnoi, in Proceedings of the 20th Annual IEEE Conference on Computational Complexity, pp. 112–119 (2005).

  • Mossel E., O’Donnell R., Servedio R.A. (2004). J. Compu. Syst. Sci. 69, 421–434

    Article  MATH  MathSciNet  Google Scholar 

  • J. Köbler and W. Lindner, Bull. EATCS 89 (2006).

  • Y. Mansour, Learning Boolean functions via the Fourier transform, in Theoretical Advances in Neural Computation and Learning (Kluwer Academic Publishers, 1994, pp. 391–424).

  • Kushilevitz E., Mansour Y. (1993). SIAM J. Comput. 22, 1331–1348

    Article  MATH  MathSciNet  Google Scholar 

  • Goldreich O., Goldwasser S., Ron D. (1998). J. ACM 45, 653–750

    Article  MATH  MathSciNet  Google Scholar 

  • Rubinfeld R., Sudan M. (1996). SIAM J. Comp. 25, 252–271

    Article  MATH  MathSciNet  Google Scholar 

  • Kahn J., Kalai G., and Linial N., in Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, pp. 68–80 (1988).

  • Bernstein E., Vazirani U. (1997). SIAM J. Comp 26, 1411–1473

    Article  MATH  MathSciNet  Google Scholar 

  • R. O’Donnell and R. A. Servedio, J. Comp System Sci, to appear. Available at http://www.cs.columbia.edu/rocco/papers/ccc03.htm.. Preliminary version appeared in Eighteenth Annual IEEE Conference on Computational Complexity, pp. 3–12 (2003).

  • Bshouty N., Cleve R., Gavaldà R., Kannan S., and Tamon C. (1996). J. Comp. Syst. Sci. 52, 421–433

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rocco A. Servedio.

Additional information

Work done while at the Department of Mathematics, Columbia University, New York, NY 10027, USA.

Supported in part by NSF award CCF-0347282, by NSF award CCF-0523664, and by a Sloan Foundation Fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Atıcı, A., Servedio, R.A. Quantum Algorithms for Learning and Testing Juntas. Quantum Inf Process 6, 323–348 (2007). https://doi.org/10.1007/s11128-007-0061-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-007-0061-6

Keywords

PACS

Navigation