  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:
  - whose sample complexity has no dependence on n, the dimension of the domain
the Boolean functions are defined over;
  - 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;
  - 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:
  - We give an algorithm for testing k-juntas to accuracy $\epsilon$ that uses
$O(k/\epsilon)$ quantum examples. This improves on the number of examples used
by the best known classical algorithm.
  - We establish the following lower bound: any FS-based k-junta testing
algorithm requires $\Omega(\sqrt{k})$ queries.
  - We give an algorithm for learning $k$-juntas to accuracy $\epsilon$ that
uses $O(\epsilon^{-1} k\log k)$ quantum examples and $O(2^k \log(1/\epsilon))$
random examples. We show that this learning algorithms is close to optimal by
giving a related lower bound.
