Quantum Algorithms for Learning and Testing Juntas

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 ε that uses O(k/ε) 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 Ω\(sqrt{k}\) queries. - We give an algorithm for learning k-juntas to accuracy ε that uses O\(ε^-1 klog k\) quantum examples and O\(2^k log(1/ε\)) random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.