Quantum Algorithms for Learning and Testing Juntas

Abstract

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: (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 uniformly 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 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: (1) 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. (2) We establish the following lower bound: any FS-based k-junta testing algorithm requires $$\Omega(\sqrt{k})$$ queries. (3) We give an algorithm for learning k-juntas to accuracy ϵ that uses O(ϵ−1 k log k) quantum examples and O(2 k log(1/ϵ)) random examples. We show that this learning algorithm is close to optimal by giving a related lower bound.