On the Walsh and Fourier-Hadamard Supports of Boolean Functions From a Quantum Viewpoint

TL;DR

This paper explores the Fourier-Hadamard and Walsh supports of Boolean functions from a quantum perspective, introducing new concepts like fully-balanced functions and analyzing their properties and applications in quantum algorithms.

Contribution

It introduces the notion of fully-balanced functions, analyzes their Fourier-Hadamard and Walsh supports, and extends results to pseudo-Boolean functions with quantum algorithm implications.

Findings

Characterization of fully-balanced functions

Analysis of supports using balancing sets

Extension to pseudo-Boolean functions and quantum applications

Abstract

In this paper, we focus on the links between Boolean function theory and quantum computing. In particular, we study the notion of what we call fully-balanced functions and analyse the Fourier--Hadamard and Walsh supports of those functions having such property. We study the Walsh and Fourier supports of other relevant classes of functions, using what we call balancing sets. This leads us to revisit and complete certain classic results and to propose new ones. We complete our study by extending the previous results to pseudo-Boolean functions (in relation to vectorial functions) and giving an insight on its applications in the analysis of the possibilities that a certain family of quantum algorithms can offer.