Quantum algorithms for the Goldreich–Levin learning problem

Abstract

The Goldreich–Levin algorithm was originally proposed for a cryptographic purpose and then applied to learning. The algorithm is to find some larger Walsh coefficients of an n variable Boolean function. Roughly speaking, it takes a \(poly(n,\frac{1}{\epsilon }\log \frac{1}{\delta })\) time to output the vectors w with Walsh coefficients \(S(w)\ge \epsilon \) with probability at least \(1-\delta \). However, in this paper, a quantum algorithm for this problem is given with query complexity \(O(\frac{\log \frac{1}{\delta }}{\epsilon ^4})\), which is independent of n. Furthermore, the quantum algorithm is generalized to apply for an n variable m output Boolean function F with query complexity \(O(2^m\frac{\log \frac{1}{\delta }}{\epsilon ^4})\).