Refined Belief-Propagation Decoding of Quantum Codes with Scalar Messages

TL;DR

This paper introduces a scalar message belief-propagation decoding algorithm for quantum codes that reduces complexity and improves performance by using message normalization and optimized update schedules.

Contribution

It proposes a refined BP algorithm for quantum codes that maintains decoding accuracy while lowering check-node complexity to binary levels and enhancing performance with message normalization.

Findings

Scalar BP achieves same decoding as quaternary BP with lower complexity

Message normalization improves decoding performance and error-floor

Serial scheduling enhances BP effectiveness against short cycles

Abstract

Codes based on sparse matrices have good performance and can be efficiently decoded by belief-propagation (BP). Decoding binary stabilizer codes needs a quaternary BP for (additive) codes over GF(4), which has a higher check-node complexity compared to a binary BP for codes over GF(2). Moreover, BP decoding of stabilizer codes suffers a performance loss from the short cycles in the underlying Tanner graph. In this paper, we propose a refined BP algorithm for decoding quantum codes by passing scalar messages. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but with a check-node complexity the same as binary BP. As every message is a scalar, the message normalization can be naturally applied to improve the performance. Another observation is that the message-update schedule affects the BP decoding performance against short cycles. We show that running BP with message normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error-floor in computer simulation.