I will give a brief introduction to the problem of estimating the convergence time a quantum channel to its fixed point(s).
In particular, I will show that bounding the gap of the channel is sometimes insufficient for bounding the convergence time.
I will review some of the tools which are available (and some which will soon be), and discuss some of the difficulties
in extending classical mixing time methods to the quantum setting.
Finally, I will provide some applications of these methods introduced, and give an outlook on some open problems.