# Quantum Algorithms for Learning and Testing Juntas

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11128-007-0061-6

- Cite this article as:
- Atıcı, A. & Servedio, R.A. Quantum Inf Process (2007) 6: 323. doi:10.1007/s11128-007-0061-6

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.