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

In this paper, we present a much simpler, more direct, and more elegant approach to the equivalence problem for {\it measure-many one-way quantum finite automata} (MM-1QFAs). The approach is essentially a generalization of the work of Carlyle [J.~Math.~Anal.~Appl.~7 (1963) 167–175]. Specifically, we reduce the equivalence problem for MM-1QFAs to the equivalence of two (initial) vectors. As an application of this approach, we use it to solve the equivalence problem for {\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 $\mathrm{\Sigma }$ are equivalent 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.