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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
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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

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Footnotes

†

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

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