Another approach to the equivalence of measure-many one-way quantum finite automata and its application

TL;DR

This paper introduces a simplified, elegant method for determining the equivalence of measure-many one-way quantum finite automata (MM-1QFAs) and applies it to enhanced quantum automata, providing clear criteria for their equivalence.

Contribution

It presents a new direct approach to the equivalence problem of MM-1QFAs, reducing it to vector comparisons, and extends this method to E-1QFAs, establishing a concrete equivalence criterion.

Findings

The equivalence of two E-1QFAs can be decided by checking a finite number of vector equivalences.

The new approach simplifies the equivalence problem for MM-1QFAs.

A specific bound for the equivalence of E-1QFAs is established based on their number of states.

Abstract

In this paper, we present a much simpler, direct and elegant approach to the equivalence problem of {\it measure many one-way quantum finite automata} (MM-1QFAs). The approach is essentially generalized from the work of Carlyle [J. Math. Anal. Appl. 7 (1963) 167-175]. Namely, we reduce the equivalence problem of MM-1QFAs to that of two (initial) vectors. As an application of the approach, we utilize it to address the equivalence problem of {\it Enhanced one-way quantum finite automata} (E-1QFAs) introduced by Nayak [Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, 1999, pp.~369-376]. We prove that two E-1QFAs $\mathcal{A}_1$ and $\mathcal{A}_2$ over $\Sigma$ are equivalence if and only if they are $n_1^2+n_2^2-1$-equivalent where $n_1$ and $n_2$ are the numbers of states in $\mathcal{A}_1$ and $\mathcal{A}_2$, respectively.