  We introduce and study time-inhomogeneous quantum Markov chains with
parameter $\zeta \ge 0$ and decoherence parameter $0 \leq p \leq 1$ on finite
spaces and their large scale equilibrium properties. Here $\zeta$ resembles the
inverse temperature in the annealing random process and $p$ is the decoherence
strength of the quantum system. Numerical evaluations show that if $ \zeta$ is
small, then quantum Markov chain is ergodic for all $0 < p \le 1$ and if $
\zeta $ is large, then it has multiple limiting distributions for all $0 < p
\le 1$. In this paper, we prove the ergodic property in the high temperature
region $0 \le \zeta \le 1$. We expect that the phase transition occurs at the
critical point $\zeta_c=1$. For coherence case $p=0$, a critical behavior of
periodicity also appears at critical point $\zeta_o=2$.
