The concepts of quantum Müller automaton and quantum infinite regular language are introduced. In virtue of the fact that the image set of a quantum infinite regular language recognized by an arbitrary quantum Müller automaton is always finite and by means of semantic analysis and quantum state construction,the algebraic characterization of quantum Müller automaton is studied. It is shown that an arbitrary quantum Müller automaton is equivalent to the one which has crisp initial states and transition relation but with quantum final states.Based on this, the algebraic description and the level characterization of quantum infinite regular languages are obtained, and it is proved that an arbitrary quantum infinite language A is recognizable if and only if the image set of A is finite and A can be represented as a finite union of special quantum infinite languages. As applications, it is proved that even if the distributivity law does not hold in quantum logic itself, quantum infinite regular languages are still closed under regular operations.

quantum logic; quantum Müller automaton; quantum infinite regular language