Quantum algorithms for the Goldreich-Levin learning problem

TL;DR

This paper presents a quantum algorithm for the Goldreich-Levin problem that significantly reduces query complexity and extends to functions with multiple outputs, outperforming classical methods.

Contribution

The paper introduces a quantum algorithm with query complexity independent of input size for finding large Walsh coefficients, and generalizes it to multi-output Boolean functions.

Findings

Quantum algorithm has query complexity $O(rac{ ext{log}(1/\delta)}{\epsilon^4})$

Algorithm's complexity is independent of input size $n$

Extended to multi-output functions with complexity $O(2^mrac{ ext{log}(1/\delta)}{\epsilon^4})$

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 $S(w)\geq\epsilon$ 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 $O(2^m\frac{\log\frac{1}{\delta}}{\epsilon^4})$.