  Motivated by a recent use of Glauber dynamics for Monte-Carlo simulations of
path integral representation of quantum spin models [Krzakala, Rosso,
Semerjian, and Zamponi, Phys. Rev. B (2008)], we analyse a natural Glauber
dynamics for the quantum Ising model with a transverse field on a finite graph
$G$. We establish strict monotonicity properties of the equilibrium
distribution and we extend (and improve) the censoring inequality of Peres and
Winkler 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 as
the "cavity equation") together with sharp bounds on the operator norm of the
derivative at the stable fixed point. It is here that the main difference
between the quantum and the classical case appear, as the cavity equation is
formulated here in an infinite dimensional vector space, whereas in the
classical case marginals belong to a one-dimensional space.
