Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T09:15:15.983Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum algorithms for testing and learning Boolean functions†

Published online by Cambridge University Press:  28 February 2013

DOMINIK FLOESS ,
ERIKA ANDERSSON  and
MARK HILLERY
DOMINIK FLOESS
Affiliation:
SUPA, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom Email: d.floess@pi4.uni-stuttgart.de; E.Andersson@hw.ac.uk
ERIKA ANDERSSON
Affiliation:
SUPA, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom Email: d.floess@pi4.uni-stuttgart.de; E.Andersson@hw.ac.uk
MARK HILLERY
Affiliation:
Department of Physics, Hunter College of CUNY, Park Avenue, New York, NY 10061, U.S.A. Email: mhillery@hunter.cuny.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss quantum algorithms based on the Bernstein–Vazirani algorithm for finding which input variables a Boolean function depends on. There are 2n possible linear Boolean functions of n input variables; given a linear Boolean function, the Bernstein–Vazirani quantum algorithm can deterministically identify which one of these Boolean functions we are given using just one single function query. We show how the same quantum algorithm can also be used to learn which input variables any other type of Boolean function} depends on. The success probability of learning that the function depends on a particular input variable depends on} the form of the Boolean function that is tested, but does not depend on the total number of input variables. We also outline a procedure based on another quantum algorithm, the Grover search, to amplify further the success probability. Finally, we discuss quantum algorithms for learning the exact form of certain quadratic and cubic Boolean functions.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 23 , Special Issue 2: Developments In Computational Models 2010 , April 2013 , pp. 386 - 398
Copyright
Copyright © Cambridge University Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

This work was partially supported by the National Science Foundation under grant PHY-0903660 and by EPSRC grant EP/G009821/1.

References

Ambainis, A. (2002) Quantum lower bounds by quantum arguments. Journal of Computer and System Science 64 750–76.CrossRefGoogle Scholar
Ambainis, A., Iwama, K., Kawachi, A., Masuda, H., Putra, R. H. and Yamashita, S. (2004) Quantum Identification of Boolean Oracles. In: Diekert, V. and Habib, M. (eds.) Proceedings of STACS 2004. Springer-Verlag Lecture Notes in Computer Science 2996 105–116.CrossRefGoogle Scholar
Atici, A. and Serviedo, R. A. (2007) Quantum Algorithms for Learning and Testing Juntas. Quantum Information Processing 6 323–348.CrossRefGoogle Scholar
Bernstein, E. and Vazirani, U. (1993) Quantum Complexity Theory. In: Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, ACM Press 11–20.Google Scholar
Brassard, G., Hoyer, P., Boyer, M. and Tapp, A. (1998) Tight bounds on quantum searching. Fortschritte der Physik 46 493–505.Google Scholar
Cleve, R., Ekert, A., Macchiavello, C. and Mosca, M. (1998) Quantum Algorithms Revisited. Proceedings of the Royal Society of London, Series A 454 339–354.CrossRefGoogle Scholar
Grover, L. (1997) Quantum Mechanics Helps in Searching for a Needle in a Haystack. Physics Review Letters 79 325–328.CrossRefGoogle Scholar
Floess, D. F. (2010) Quantum Mechanics and Boolean Functions, Master's thesis, Heriot-Watt University.Google Scholar
Iwama, K., Kawachi, A., Masuda, H., Putra, R. H. and Yamashita, S. (2003) Quantum Evaluation of Multi-Valued Boolean Functions. (Available at quant-ph/0304131.)Google Scholar
Rötteler, M. (2009) Quantum algorithms to solve the hidden shift problem for quadratics and for functions of large Gowers norm. In: Krlovic, R. and Niwinski, D. (eds.) Mathematical Foundations of Computer Science 2009, Proceedings. Springer-Verlag Lecture Notes in Computer Science 5734 663–674.CrossRefGoogle Scholar