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

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