  We provide a rigorous proof of the approximate convergence of sliding-window
quantum belief-propagation as outlined heuristically in the work of Bilgin and
Poulin (Ref. [1]), in the absence of the quantum Markov property. In
particular, we confirm the hypothesis outlined in this work that the
approximation error of each step in the belief-propagation algorithm decreases
exponentially with the sliding-window size, under the assumption that the
underlying state on which belief-propagation is being performed possesses a
so-called thermal boundedness property: a relaxation of the Markov property
required for exact convergence.
