  {\it Two-way finite automata with quantum and classical states} (2QCFA) were
introduced by Ambainis and Watrous, and it was shown that 2QCFA have
superiority over {\it two-way probabilistic finite automata} (2PFA) for
recognizing some non-regular languages such as the language
$L_{eq}=\{a^{n}b^{n}\mid n\in \mathbf{N}\}$ and the palindrome language
$L_{pal}=\{\omega\in \{a,b\}^*\mid\omega=\omega^R\}$, where $x^R$ is $x$ in the
reverse order. It is interesting to find more languages like these that witness
the superiority of 2QCFA over 2PFA. In this paper, we consider the language
$L_{m}=\{xcy\mid \Sigma=\{a, b, c\}, x,y\in\{a,b\}^{*},c\in\Sigma, |x|=|y|\}$
that is similar to the middle language $L_{middle}=\{xay\mid
x,y\in\Sigma^{*},a\in\Sigma, |x|=|y|\}$. We prove that the language $L_{m}$ can
be recognized by 2QCFA with one-sided error in polynomial expected time. Also,
we show that $L_{m}$ can be recognized by 2PFA with bounded error, but only in
exponential expected time. Thus $L_{m}$ is another witness of the fact that
2QCFA are more powerful than their classical counterparts.
