Quantum boolean functions

TL;DR

This paper introduces quantum boolean functions, extending classical boolean function analysis to the quantum domain, including quantum property testing, Fourier analysis, and hypercontractive inequalities, advancing quantum computational theory.

Contribution

It presents the first systematic study of quantum boolean functions, generalizing key classical results and developing new quantum algorithms and inequalities.

Findings

Quantum property testing methods for boolean functions.

A quantum version of the Goldreich-Levin algorithm.

Quantum hypercontractive inequality extension.

Abstract

In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f^2 = I. We describe several generalisations of well-known results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich-Levin algorithm for finding the large Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. In order to obtain one of these generalisations, we prove a quantum extension of the hypercontractive inequality of Bonami, Gross and Beckner.