Periodic Fourier representation of Boolean functions

TL;DR

This paper introduces the periodic Fourier representation of Boolean functions, linking it to quantum computation models and revealing how the complexity of functions relates to their Fourier and polynomial degrees.

Contribution

It defines the periodic Fourier representation, explores its properties, and establishes bounds connecting it to the $ ext{F}_2$-degree and quantum computational complexity.

Findings

Boolean functions related to $ ext{Z}/4 ext{Z}$-polynomials have small periodic Fourier sparsities.

Periodic Fourier sparsity is at least $2^{ ext{deg}_{ ext{F}_2}(f)} - 1$, linking complexity to $ ext{F}_2$-degree.

Symmetric Boolean functions can be computed efficiently by depth-2 $ ext{NMQC}_igoplus$, showing exponential gaps in quantum models.

Abstract

In this work, we consider a new type of Fourier-like representation of Boolean function $f\colon\{+1,-1\}^n\to\{+1,-1\}$ \[ f(x) = \cos\left(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i\right). \] This representation, which we call the periodic Fourier representation, of Boolean function is closely related to a certain type of multipartite Bell inequalities and non-adaptive measurement-based quantum computation with linear side-processing ($\mathrm{NMQC}_\oplus$). The minimum number of non-zero coefficients in the above representation, which we call the periodic Fourier sparsity, is equal to the required number of qubits for the exact computation of $f$ by $\mathrm{NMQC}_\oplus$. Periodic Fourier representations are not unique, and can be directly obtained both from the Fourier representation and the $\mathbb{F}_2$-polynomial representation. In this work, we first show that Boolean functions related to $\mathbb{Z}/4\mathbb{Z}$-polynomial have small periodic Fourier sparsities. Second, we show that the periodic Fourier sparsity is at least $2^{\mathrm{deg}_{\mathbb{F}_2}(f)}-1$, which means that $\mathrm{NMQC}_\oplus$ efficiently computes a Boolean function $f$ if and only if $\mathbb{F}_2$-degree of $f$ is small. Furthermore, we show that any symmetric Boolean function, e.g., $\mathsf{AND}_n$, $\mathsf{Mod}^3_n$, $\mathsf{Maj}_n$, etc, can be exactly computed by depth-2 $\mathrm{NMQC}_\oplus$ using a polynomial number of qubits, that implies exponential gaps between $\mathrm{NMQC}_\oplus$ and depth-2 $\mathrm{NMQC}_\oplus$.