Polar Codes for CQ Channels: Decoding via Belief-Propagation with Quantum Messages

TL;DR

This paper introduces a belief-propagation decoding method for polar codes over classical-quantum channels, enabling efficient code design via density evolution and demonstrating theoretical validation despite classical simulation limitations.

Contribution

It proposes a novel paired-measurement BPQM decoding scheme for polar codes on CQ channels, allowing classical density evolution analysis for code design.

Findings

Density evolution analysis for PM-BPQM decoding

Efficient code design for CQ channels using DE

Validation of theoretical results through limited classical simulations

Abstract

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.