# Modified belief propagation decoders for quantum low-density parity-check codes

## TL;DR

This paper introduces modified belief propagation decoders for quantum LDPC codes that improve decoding performance by addressing issues like 4-cycles and error correlations, outperforming existing methods.

## Contribution

The paper proposes novel belief propagation decoders for quantum LDPC codes, including error correlation reintroduction and augmented decoding techniques, enhancing decoding accuracy.

## Key findings

- Decoders outperform existing methods in various quantum LDPC codes.
- Reintroducing error correlations improves decoding success.
- Augmented decoders with duplicated check nodes enhance performance.

## Abstract

Quantum low-density parity-check codes can be decoded using a syndrome based $\mathrm{GF}(4)$ belief propagation decoder. However, the performance of this decoder is limited both by unavoidable $4$-cycles in the code's factor graph and the degenerate nature of quantum errors. For the subclass of CSS codes, the number of $4$-cycles can be reduced by breaking an error into an $X$ and $Z$ component and decoding each with an individual $\mathrm{GF}(2)$ based decoder. However, this comes at the expense of ignoring potential correlations between these two error components. We present a number of modified belief propagation decoders that address these issues. We propose a $\mathrm{GF}(2)$ based decoder for CSS codes that reintroduces error correlations by reattempting decoding with adjusted error probabilities. We also propose the use of an augmented decoder, which has previously been suggested for classical binary low-density parity-check codes. This decoder iteratively reattempts decoding on factor graphs that have a subset of their check nodes duplicated. The augmented decoder can be based on a $\mathrm{GF}(4)$ decoder for any code, a $\mathrm{GF}(2)$ decoder for CSS code, or even a supernode decoder for a dual-containing CSS code. For CSS codes, we further propose a $\mathrm{GF}(2)$ based decoder that combines the augmented decoder with error probability adjustment. We demonstrate the performance of these new decoders on a range of different codes, showing that they perform favorably compared to other decoders presented in literature.