State complexity of one-way quantum finite automata together with classical states

One-way quantum finite automata together with classical states (1QFAC) proposed by Qiu et al. is a new one-way quantum finite automata (1QFA) model that integrates quantum finite automata (QFA) and deterministic finite automata (DFA). The relationshipsand balances between quantum states and classical states in 1QFAC are still not clear. In this paper, we obtain the following results: (1) We optimize the bound given by Qiu et al. that characterizes the relationships between quantum basis states and classical states as well as the equivalent minimal DFA. (2) We give an upper bound showing how many classical states are needed upon reducing the quantum basis states of 1QFAC. (3) We give a lower bound on the classical state number of 1QFAC for recognizing any given regular language, and show that the lower bound is exact if the given language is finite. (4) We show that 1QFAC are exponentially more succinct than DFA and probabilistic finite automata (PFA) for some regular languages. (5) We point out essential relationships between 1QFAC, MO-1QFA and multi-letter 1QFA, and induce a result regarding quantitative relationships between multi-letter 1QFA and DFA.