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Quaternary Neural Belief Propagation Decoding of Quantum LDPC Codes with Overcomplete Check Matrices

Sisi Miao, Alexander Schnerring, Haizheng Li, Laurent Schmalen

TL;DR

This work targets universal, low-latency decoding for quantum LDPC codes by combining belief propagation on overcomplete check matrices with neural enhancements. It develops a quaternary BP4 decoder and a degeneracy-aware neural BP (NBP4/NOBP4) framework, including an OC matrix construction strategy that augments the Tanner graph with redundant checks. Empirical results show large FER reductions across GB and toric codes, with OBP4 substantially improving convergence and NOBP4 exploiting code degeneracy to approach or surpass MWPM in several regimes. The approach offers practical pathways to fast, robust quantum error correction, and the released code enables replication and further exploration.

Abstract

Quantum low-density parity-check (QLDPC) codes are promising candidates for error correction in quantum computers. One of the major challenges in implementing QLDPC codes in quantum computers is the lack of a universal decoder. In this work, we first propose to decode QLDPC codes with a belief propagation (BP) decoder operating on overcomplete check matrices. Then, we extend the neural BP (NBP) decoder, which was originally studied for suboptimal binary BP decoding of QLPDC codes, to quaternary BP decoders. Numerical simulation results demonstrate that both approaches as well as their combination yield a low-latency, high-performance decoder for several short to moderate length QLDPC codes.

Quaternary Neural Belief Propagation Decoding of Quantum LDPC Codes with Overcomplete Check Matrices

TL;DR

This work targets universal, low-latency decoding for quantum LDPC codes by combining belief propagation on overcomplete check matrices with neural enhancements. It develops a quaternary BP4 decoder and a degeneracy-aware neural BP (NBP4/NOBP4) framework, including an OC matrix construction strategy that augments the Tanner graph with redundant checks. Empirical results show large FER reductions across GB and toric codes, with OBP4 substantially improving convergence and NOBP4 exploiting code degeneracy to approach or surpass MWPM in several regimes. The approach offers practical pathways to fast, robust quantum error correction, and the released code enables replication and further exploration.

Abstract

Quantum low-density parity-check (QLDPC) codes are promising candidates for error correction in quantum computers. One of the major challenges in implementing QLDPC codes in quantum computers is the lack of a universal decoder. In this work, we first propose to decode QLDPC codes with a belief propagation (BP) decoder operating on overcomplete check matrices. Then, we extend the neural BP (NBP) decoder, which was originally studied for suboptimal binary BP decoding of QLPDC codes, to quaternary BP decoders. Numerical simulation results demonstrate that both approaches as well as their combination yield a low-latency, high-performance decoder for several short to moderate length QLDPC codes.
Paper Structure (27 sections, 21 equations, 11 figures, 3 tables)

This paper contains 27 sections, 21 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stabilizer Codes
  4. Codes Used in this Study
  5. Error Correction with Stabilizer Codes
  6. Belief Propagation Decoder
  7. Refined BP4 Decoder
  8. Initialization of the BP Decoder
  9. Overcomplete Check Matrices
  10. Constructing Redundant Checks
  11. Heuristic Analysis
  12. Neural Belief Propagation
  13. Framework
  14. Loss Function
  15. Initial Weights of the Redundant Checks
  16. ...and 12 more sections

Figures (11)

  • Figure 1: Block diagramm of QEC using QSC.
  • Figure 2: Tanner graph of the $[[7,1,3]]$ quantum BCH code, with VNs represented by circles and CNs by squares. Blue solid edges represent coefficients $\omega$ ($\bm{X}$ type), while yellow dashed edges represent coefficients $\overline{\omega}$ ($\bm{Z}$ type).
  • Figure 3: FER under different initial $\epsilon_0$ values: Left figure: toric code ($d=8$), middle figure: GB-A1 code, right figure: GB-A2 code.
  • Figure 4: Example of the construction of a weight-6 stabilizer for the toric code with $d=4$. The red edges are the qubits which support an $X$ stabilizer.
  • Figure 5: BP decoding results for the [[7,1,3]] CSS code with original and overcomplete check matrix ($L=32$).
  • ...and 6 more figures