Quantum mixing of Markov chains for special distributions

The preparation of the stationary distribution of irreducible, time-reversible Markov chains is a fundamental building block in many heuristic approaches to algorithmically hard problems. It has been conjectured that quantum analogs of classical mixing processes may offer a generic quadratic speed-up in realizing such stationary distributions. Such a speed-up would also imply a speed-up of a broad family of heuristic algorithms. However, a true quadratic speed up has thus far only been demonstrated for special classes of Markov chains. These results often presuppose a regular structure of the underlying graph of the Markov chain, and also a regularity in the transition probabilities. In this work, we demonstrate a true quadratic speed-up for a class of Markov chains where the restriction is only on the form of the stationary distribution, rather than directly on the Markov chain structure itself. In particular, we show efficient mixing can be achieved when it is beforehand known that the distribution is monotonically decreasing relative to a known order on the state space. Following this, we show that our approach extends to a wider class of distributions, where only a fraction of the shape of the distribution is known to be monotonic. Our approach is built on the Szegedy-type quantization of transition operators.