  This paper considers the design and decoding of polar codes for general
classical-quantum (CQ) channels. It focuses on decoding via belief-propagation
with quantum messages (BPQM) and, in particular, the idea of paired-measurement
BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density
evolution (DE) analysis, one can use DE to design a polar code for any CQ
channel and then efficiently compute the trade-off between code rate and error
probability. We have also implemented and tested a classical simulation of our
PM-BPQM decoder for polar codes. While the decoder can be implemented
efficiently on a quantum computer, simulating the decoder on a classical
computer actually has exponential complexity. Thus, simulation results for the
decoder are somewhat limited and are included primarily to validate our
theoretical results.
