One-way finite automata with quantum and classical states

In this paper, we introduce and explore a new model of {\it quantum finite automata} (QFA). Namely, {\it one-way finite automata with quantum and classical states} (1QCFA), a one way version of {\it two-way finite automata with quantum and classical states} (2QCFA) introduced by Ambainis and Watrous in 2002 \cite{AJ}. First, we prove that {\it one-way probabilistic finite automata} (1PFA) \cite{AP} and {\it one-way quantum finite automata with control language} (1QFACL) \cite{ACB} as well as several other models of QFA, can be simulated by 1QCFA. Afterwards, we explore several closure properties for the family of languages accepted by 1QCFA. Finally, the state complexity of 1QCFA is explored and the main succinctness result is derived. Namely, for any prime $m$ and any $\epsilon_1>0$, there exists a language $L_{m}$ that cannot be recognized by any {\it measure-many one-way quantum finite automata} (MM-1QFA) \cite{Kon97} with bounded error $7/9+\epsilon_1$, and any 1PFA recognizing it has at last $m$ states, but $L_{m}$ can be recognized by a 1QCFA for any error bound $\epsilon>0$ with $\bf{O}(\log{m})$ quantum states and 12 classical states.