Sequential quantum mixing for slowly evolving sequences of Markov chains

In this work we consider the problem of preparation of the stationary distribution of irreducible, time-reversible Markov chains, which is a fundamental task in algorithmic Markov chain theory. For the classical setting, this task has a complexity lower bound of $\Omega(1/\delta)$, where $\delta$ is the spectral gap of the Markov chain, and other dependencies contribute only logarithmically. In the quantum case, the conjectured complexity is $O(\sqrt{\delta^{-1}})$ (with other dependencies contributing only logarithmically). However, this bound has only been achieved for a few special classes of Markov chains. In this work, we provide a method for the sequential preparation of stationary distributions for sequences of general time-reversible $N-$state Markov chains, akin to the setting of simulated annealing methods. The complexity of preparation we achieve is $O(\sqrt{\delta^{-1}} N^{1/4})$, neglecting logarithmic factors. While this result falls short of the conjectured optimal time, it still provides at least a quadratic improvement over other straightforward approaches for quantum mixing applied in this setting.