Beyond z=0. The Deutsch-Jozsa decided monochromatic languages

TL;DR

This paper explores the class of languages for which the Deutsch-Jozsa algorithm outputs monochromatic results beyond trivial cases, using classical simulations to analyze their properties and implications for quantum decision processes.

Contribution

It introduces the concept of Deutsch-Jozsa decided languages beyond trivial cases and demonstrates their properties through classical simulations for specific qubit counts.

Findings

Monochromatic cases extend beyond trivial constant functions.

Simulations suggest these languages are equivalent to superpositions of monochromatic cases.

Provides insights into quantum decision boundaries for language classes.

Abstract

The present work points out that the Deutsch-Jozsa algorithm was the first formal description of a quantum decider. In particular, it is studied here the class of languages whose indicator functions allow the Deutsch-Jozsa algorithm to output a monochromatic result, beyond the trivial case z = 0 for constant indicator functions. To illustrate examples of randomly balanced languages and some monochromatic cases, it was performed classical computational simulations of the Deutsch-Jozsa quantum algorithm for the specific cases of 4 and 6 qubits, respectively. The general case of the Deutsch-Jozsa decided languages are named balanced languages, and their outcomes from the simulation suggest that such languages are equivalent to the quantum superposition of the monochromatic cases.