Regular quantum grammars and regular grammars

To study quantum computation, Moore and Crutchfield (Theoret. Comput. Sci. vol. 237, pp. 275-306, 2000) proposed quantum versions of finite-state and push-down automata, as well as regular and context-free grammars. In this note, we show that the languages generated by regular quantum grammars are equivalent to the languages generated by regular grammars. Therefore, from the perspective of generating languages, regular quantum grammars are not more powerful than classical regular grammars. This implies that it may be necessary to look for more well-formed definitions of quantum grammars and maybe those of quantum automata.