Quantum Information Processing

, Volume 6, Issue 5, pp 323–348

Quantum Algorithms for Learning and Testing Juntas

Article

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−1k log k) quantum examples and O(2k log(1/ϵ)) random examples. We show that this learning algorithm is close to optimal by giving a related lower bound.

Keywords

Juntasquantum query algorithmsquantum property testingcomputational learning theoryquantum computationlower bounds

PACS

03.67.-a03.67.Lx

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Citadel Investment GroupChicagoUSA
  2. 2.Department of Computer ScienceColumbia UniversityNew YorkUSA