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Motivated by a recent use of Glauber dynamics for Monte-Carlo simulations ofpath integral representation of quantum spin models [Krzakala, Rosso,Semerjian, and Zamponi, Phys. Rev. B (2008)], we analyse a natural Glauberdynamics for the quantum Ising model with a transverse field on a finite graph$G$. We establish strict monotonicity properties of the equilibriumdistribution and we extend (and improve) the censoring inequality of Peres andWinkler to the quantum setting. Then we consider the case when $G$ is a regular$b$-ary tree and prove the same fast mixing results established in [Martinelli,Sinclair, and Weitz, Comm. Math. Phys. (2004)] for the classical Ising model.Our main tool is an inductive relation between conditional marginals (known asthe "cavity equation") together with sharp bounds on the operator norm of thederivative at the stable fixed point. It is here that the main differencebetween the quantum and the classical case appear, as the cavity equation isformulated here in an infinite dimensional vector space, whereas in theclassical case marginals belong to a one-dimensional space.

Glauber dynamics for the quantum Ising model in a transverse field on a regular tree