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

摘要

In this paper, we present a much simpler, direct and elegant approach to the equivalence problem of measure many one-way quantum finite automata (MM-1QFAs). The approach is essentially generalized from the work of Carlyle [J.W. Carlyle, Reduced forms for stochastic sequential machines, 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 enhanced one-way quantum finite automata (E-1QFAs) introduced by Nayak [A. Nayak, Optimal lower bounds for quantum automata and random access codes, in: Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), 1999, pp. 369–376]. We prove that two E-1QFAs A1 and A2 over Σ are equivalence if and only if they are n12+n22−1-equivalent where n1 and n2 are the numbers of states in A1 and A2, respectively. As an important consequence, we obtain that it is decidable whether or not L(A1)=L(A2) where ⁎L(A)⊆Σ⁎ denotes the set recognizable by MM-1QFA A (or by E-1QFA A) with cutpoint (or with non-strict cutpoint). This also extends a theorem of Eilenberg.