Algebraic Characterization of the Class of Languages recognized by Measure Only Quantum Automata

TL;DR

This paper characterizes the class of languages recognized by measure-only quantum automata using algebraic structures, establishing their equivalence with certain idempotent monoid varieties and providing a polynomial-time decision algorithm.

Contribution

It offers an algebraic characterization of measure-only quantum automata languages and connects them with well-known algebraic varieties, enabling efficient membership testing.

Findings

Languages in LMO have syntactic monoids that are J-trivial and idempotent.

LMO coincides with literally idempotent piecewise testable regular languages.

A polynomial-time algorithm exists to decide language membership in LMO.

Abstract

We study a model of one-way quantum automaton where only measurement operations are allowed (MOn-1qfa). We give an algebraic characterization of LMO, showing that the syntactic monoids of the languages in LMO are exactly the literal pseudovariety of J-trivial literally idempotent monoids, where J is the Green's relation determined by two-sided ideals. We also prove that LMO coincides with the literal variety of literally idempotent piecewise testable regular languages. This allows us to prove the existence of a polynomial time algorithm for deciding whether a regular language belongs to LMO.