  Markov chain methods are remarkably successful in computational physics,
machine learning, and combinatorial optimization. The cost of such methods
often reduces to the mixing time, i.e., the time required to reach the steady
state of the Markov chain, which scales as $\delta^{-1}$, the inverse of the
spectral gap. It has long been conjectured that quantum computers offer nearly
generic quadratic improvements for mixing problems. However, except in special
cases, quantum algorithms achieve a run-time of $\mathcal{O}(\sqrt{\delta^{-1}}
\sqrt{N})$, which introduces a costly dependence on the Markov chain size $N,$
not present in the classical case. Here, we re-address the problem of mixing of
Markov chains when these form a slowly evolving sequence. This setting is akin
to the simulated annealing setting and is commonly encountered in physics,
material sciences and machine learning. We provide a quantum memory-efficient
algorithm with a run-time of $\mathcal{O}(\sqrt{\delta^{-1}} \sqrt[4]{N})$,
neglecting logarithmic terms, which is an important improvement for large state
spaces. Moreover, our algorithms output quantum encodings of distributions,
which has advantages over classical outputs. Finally, we discuss the run-time
bounds of mixing algorithms and show that, under certain assumptions, our
algorithms are optimal.
