An application of the Deutsch-Josza algorithm to formal languages and the word problem in groups

TL;DR

This paper adapts the Deutsch-Josza quantum algorithm to formal language theory, enabling efficient distinction of trivial and nontrivial words in groups, thus impacting the word problem in group theory.

Contribution

It extends the Deutsch-Josza algorithm to functions with arbitrary binary output and applies it to solve specific cases of the word problem in groups.

Findings

Reduced number of oracle queries compared to classical methods

Demonstrated application to group word problem

Extended algorithm to functions with arbitrary length binary output

Abstract

We adapt the Deutsch-Josza algorithm to the context of formal language theory. Specifically, we use the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. This is done by extending the original algorithm to functions of arbitrary length binary output, with the introduction of a more general concept of parity. We provide examples in which properties of the algorithm allow to reduce the number of oracle queries with respect to the deterministic classical case. This has some consequences for the word problem in groups with a particular kind of presentation.