  Quantum speedup for mixing a Markov chain can be achieved based on the
construction of slowly-varying $r$ Markov chains where the initial chain can be
easily prepared and the spectral gaps have uniform lower bound. The overall
complexity is proportional to $r$. We present a multi-level approach to
construct such a sequence of $r$ Markov chains by varying a resolution
parameter $h.$ We show that the density function of a low-resolution Markov
chain can be used to warm start the Markov chain with high resolution. We prove
that in terms of the chain length the new algorithm has $O(1)$ complexity
rather than $O(r).$
