  One-way quantum finite automata together with classical states (1QFAC)
proposed in [Journal of Computer and System Sciences 81(2) (2015) 359--375] is
a new one-way quantum finite automata (1QFA) model that integrates quantum
finite automata (QFA) and deterministic finite automata (DFA). This model uses
classical states to control the evolution and measurement of quantum states. As
a quantum-classical hybrid model, 1QFAC recognize all regular languages. It was
shown that the state complexity of 1QFAC for some languages is essentially
superior to that of DFA and other 1QFA. In this paper, our goal is to clarify
state complexity problems for 1QFAC. We obtain the following results: (1) We
optimize the bound given by Qiu et al. that characterizes the relationship
between quantum basis state number and classical state number of 1QFAC as well
as the state number of its corresponding minimal DFA for recognizing any given
regular language. (2) We give an upper bound showing that how many classical
states are needed if the quantum basis states of 1QFAC are reduced without
changing its recognition ability. (3) We give a lower bound of the classical
state number of 1QFAC for recognizing any given regular language, and 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 recognizing some regular languages that can not be recognized by
measure-once 1QFA (MO-1QFA), measure-many 1QFA (MM-1QFA) or multi-letter 1QFA.
(5) We reveal essential relationships between 1QFAC, MO-1QFA and multi-letter
1QFA, and induce a result regarding a quantitative relationship between the
state number of multi-letter 1QFA and DFA.
