Quantum Information Processing, Vol. 6, No. 5, October 2007 (© 2007)
Quantum Algorithms for Learning and Testing Juntas
and Rocco A. Servedio
Received April 12, 2007; accepted July 24, 2007; Published online: September 2, 2007
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 vari-
ables. 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 uni-
formly 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 exam-
ples). Our quantum algorithms are based on a subroutine FS which enables sam-
pling 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 that uses
O(k/) 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
k) queries. (3) We give an algo-
rithm for learning k-juntas to accuracy that uses O(
k log k) quantum exam-
ples and O(2
log(1/)) random examples. We show that this learning algorithm
is close to optimal by giving a related lower bound.
KEY WORDS: Juntas; quantum query algorithms; quantum property testing;
computational learning theory; quantum computation; lower bounds.
PAC S : 03.67.-a; 03.67.Lx.
Citadel Investment Group, Chicago, IL 60603, USA. E-mail: firstname.lastname@example.org
Department of Computer Science, Columbia University, New York, NY 10027, USA.
To whom correspondence should be addressed.
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.
1570-0755/07/1000-0323/0 © 2007 Springer Science+Business Media, LLC