Belief propagation decoding of quantum channels by passing quantum messages

TL;DR

This paper introduces a quantum belief propagation decoding algorithm that passes quantum messages, enabling efficient and explicit decoding of classical-quantum channels, including capacity-achieving decoders for non-Pauli channels.

Contribution

It develops the first explicit quantum belief propagation decoder capable of handling non-Pauli channels and integrates it with polar codes for quantum communication.

Findings

Decoding circuits have quadratic gate complexity in blocklength.

The decoder can be adapted for polar codes on pure state channels.

Achieves capacity for non-Pauli channels with explicit decoding schemes.

Abstract

Belief propagation is a powerful tool in statistical physics, machine learning, and modern coding theory. As a decoding method, it is ubiquitous in classical error correction and has also been applied to stabilizer-based quantum error correction. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding, in some cases even up to the Shannon capacity of the channel. Here we construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with pure state outputs. This gives explicit decoding circuits whose number of gates is quadratic in the blocklength of the code. We also show that this decoder can be modified to work with polar codes for the pure state channel and as part of a polar decoder for transmitting quantum information over the amplitude damping channel. These represent the first explicit capacity-achieving decoders for non-Pauli channels.