Quantum Fast-Forwarding Markov Chains

We show that it is possible to use quantum walks to quadratically fast-forward a reversible Markov chain. More specifically, let $P$ be the transition matrix of a reversible Markov chain and let $D = \sqrt{P\circ P^T}$ be its discriminant matrix ($D=P$ if $P$ is symmetric). We construct a quantum walk algorithm that starts from an arbitrary quantum state $|v>$ and returns a quantum state (up to normalization) $\epsilon$-close to the state $D^t|v>$ with probability $\|D^t|v>\|_2^2$, requiring $O\left(\sqrt{\frac{t}{\epsilon}}\log\left(\frac{1}{\epsilon}\right)\right)$ quantum walk steps.This result further substantiates the successful interaction between Markov chains and quantum walks. It also shows that quantum walks can accelerate the intermediate dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We use the algorithm to prove how quantum walks can escape large sets quadratically faster than classical Markov chains, a problem related to the open question of hitting large sets with quantum walks, and we demonstrate an application in quantum state generation.