  Belief propagation (BP) is a classical algorithm that approximates the
marginal distribution associated with a factor graph by passing messages
between adjacent nodes in the graph. It gained popularity in the 1990's as a
powerful decoding algorithm for LDPC codes. In 2016, Renes introduced a belief
propagation with quantum messages (BPQM) and described how it could be used to
decode classical codes defined by tree factor graphs that are sent over the
classical-quantum pure-state channel. In this work, we propose an extension of
BPQM to general binary-input symmetric classical-quantum (BSCQ) channels based
on the implementation of a symmetric "paired measurement". While this new
paired-measurement BPQM (PMBPQM) approach is suboptimal in general, it provides
a concrete BPQM decoder that can be implemented with local operations.
