Non-Markov quantum belief propagation

TL;DR

This paper rigorously proves that sliding-window quantum belief propagation converges approximately and exponentially fast under thermal boundedness, a relaxed condition of the quantum Markov property, extending heuristic results.

Contribution

It provides a formal proof of convergence and error bounds for quantum belief propagation without the Markov property, under thermal boundedness.

Findings

Approximate convergence of sliding-window quantum belief propagation is proven.

Error decreases exponentially with sliding-window size.

Convergence holds under thermal boundedness, a relaxation of the quantum Markov property.

Abstract

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.