A quantum algorithm to estimate the Gowers $U_2$ norm and linearity testing of Boolean functions

TL;DR

This paper introduces quantum algorithms for estimating the Gowers $U_2$ and $U_3$ norms of Boolean functions, and for testing linearity more efficiently than classical methods, with potential applications in property testing.

Contribution

The paper presents novel quantum algorithms for estimating Gowers norms and testing linearity of Boolean functions, outperforming classical algorithms like BLR.

Findings

Quantum algorithm for Gowers $U_2$ norm estimation

Quantum linearity testing algorithm with improved performance

Outline of a quantum algorithm for Gowers $U_3$ norm estimation

Abstract

We propose a quantum algorithm to estimate the Gowers $U_2$ norm of a Boolean function, and extend it into a second algorithm to distinguish between linear Boolean functions and Boolean functions that are $\epsilon$-far from the set of linear Boolean functions, which seems to perform better than the classical BLR algorithm. Finally, we outline an algorithm to estimate Gowers $U_3$ norms of Boolean functions.