On the connection between Deutsch-Jozsa algorithm and bent functions

TL;DR

This paper reveals a connection between the Deutsch-Jozsa quantum algorithm and bent functions, showing that the algorithm's quantum circuit performs the Walsh transform, with bent functions producing a flat quantum spectrum.

Contribution

It establishes a novel link between quantum algorithms and Boolean function properties, specifically connecting the Deutsch-Jozsa algorithm to bent functions.

Findings

Quantum circuit performs Walsh transform of Boolean functions.

Bent functions yield a flat spectrum in the Deutsch-Jozsa algorithm.

Provides insight into the spectral properties of quantum states for bent functions.

Abstract

It is presently shown that the Deutsch-Jozsa algorithm is connected to the concept of bent function. Particularly, it is noticeable that the quantum circuit used to denote the well known quantum algorithm is by itself the quantum computer which performs the Walsh transform of a Boolean function. Consequently, the output from Deutsch-Jozsa algorithm, when the hidden function is bent, corresponds to a flat spectrum of quantum states.