Upper bounds on the size of one-way quantum finite automata

We show that, for any stochastic event p of period n, there exists a measure-once one-way quantum finite automaton (1qfa) with at most 2 √ 6n + 25 states inducing the event ap + b, for constants a > 0, b≥ 0, satisfying a+b ≤1. This fact is proved by designing an algorithm which constructs the desired 1qfa in polynomial time. As a consequence, we get that any periodic language of period n can be accepted with isolated cut point by a 1qfa with no more than 2 √ 6n+26 states. Our results give added evidence of the strength of measure-once 1qfa’s with respect to classical automata.