Log-space counter is useful for unary languages by help of a constant-size quantum register

The minimum amount of resources to recognize a nonregular language is a fundamental research topic in theoretical computer science which has been examined for different kinds of resources and many different models. In this note, we focus on unary languages and space complexity on counters. Our model is two-way one-counter automaton with quantum and classical states (2QCCA), which is a two-way finite automaton with one-counter (2DCA) augmented with a fixed size quantum register or a two-way finite automaton with quantum and classical states (2QCFA) augmented with a classical counter. It is known that any 2DCA using a sublinear space on its counter can recognize only regular languages \cite{DG82B}. In this note, we show that bounded-error 2QCCAs can recognize a non-regular unary language by using logarithmic space on its counters for the members. Note that it is still an open problem whether bounded-error 2QCFA can recognize a non-regular unary language.