  Random walk algorithms, including sampling and approximations, have played a
significant role in statistical physics and theoretical computer science.
Mixing through walks is the process for a Markov chain to approximate a
stationary distribution for a group. Quantum walks have shown potential
advantages in mixing time over the classical case but lack general proof in the
finite group case. Here, we investigate the continuous-time quantum walks on
Cayley graphs of the dihedral group $D_{2n}$ for odd $n$, generated by the
smallest inverse closed symmetric subset. We present a significant finding
that, in contrast to the classical mixing time on these Cayley graphs, which is
typically of order $O(n^2 \log(2n/\epsilon))$, the continuous-time quantum walk
mixing time on $D_{2n}$ is of order $O(n (\log n)^5 \log(1/\epsilon))$,
achieving a quadratic improvement over the classical case. Our paper advances
the general understanding of quantum walk mixing on Cayley graphs, highlighting
the improved mixing time achieved by continuous-time quantum walks on $D_{2n}$.
This work has potential applications in algorithms for sampling non-abelian
groups, graph isomorphism tests, etc.
