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Decoding Quantum LDPC Codes Using Graph Neural Networks

Vukan Ninkovic, Ognjen Kundacina, Dejan Vukobratovic, Christian Häger, Alexandre Graell i Amat

TL;DR

The paper tackles the challenge of decoding Quantum Low-Density Parity-Check codes, where degeneracy complicates traditional BP decoders. It introduces a Graph Neural Network (GNN) decoder that operates on the QLDPC Tanner graph to predict the binary error vector $oldsymbol{e}_{ extrm{bin}}$ from the syndrome $oldsymbol{s}$, using a two-type node, attention-based message-passing architecture with layer-specific message functions and a final predictive head trained via binary cross-entropy. The approach is evaluated on quantum hypergraph product and bicycle codes, showing substantial performance gains over conventional BP and neural-enhanced BP variants, and competitive results with neural post-processing decoders, all while maintaining linear complexity in the code length. The findings highlight the potential of GNN-based decoders to leverage graph structure and reduce decoding complexity in practical quantum error correction, supporting scalable quantum computation applications.

Abstract

In this paper, we propose a novel decoding method for Quantum Low-Density Parity-Check (QLDPC) codes based on Graph Neural Networks (GNNs). Similar to the Belief Propagation (BP)-based QLDPC decoders, the proposed GNN-based QLDPC decoder exploits the sparse graph structure of QLDPC codes and can be implemented as a message-passing decoding algorithm. We compare the proposed GNN-based decoding algorithm against selected classes of both conventional and neural-enhanced QLDPC decoding algorithms across several QLDPC code designs. The simulation results demonstrate excellent performance of GNN-based decoders along with their low complexity compared to competing methods.

Decoding Quantum LDPC Codes Using Graph Neural Networks

TL;DR

The paper tackles the challenge of decoding Quantum Low-Density Parity-Check codes, where degeneracy complicates traditional BP decoders. It introduces a Graph Neural Network (GNN) decoder that operates on the QLDPC Tanner graph to predict the binary error vector from the syndrome , using a two-type node, attention-based message-passing architecture with layer-specific message functions and a final predictive head trained via binary cross-entropy. The approach is evaluated on quantum hypergraph product and bicycle codes, showing substantial performance gains over conventional BP and neural-enhanced BP variants, and competitive results with neural post-processing decoders, all while maintaining linear complexity in the code length. The findings highlight the potential of GNN-based decoders to leverage graph structure and reduce decoding complexity in practical quantum error correction, supporting scalable quantum computation applications.

Abstract

In this paper, we propose a novel decoding method for Quantum Low-Density Parity-Check (QLDPC) codes based on Graph Neural Networks (GNNs). Similar to the Belief Propagation (BP)-based QLDPC decoders, the proposed GNN-based QLDPC decoder exploits the sparse graph structure of QLDPC codes and can be implemented as a message-passing decoding algorithm. We compare the proposed GNN-based decoding algorithm against selected classes of both conventional and neural-enhanced QLDPC decoding algorithms across several QLDPC code designs. The simulation results demonstrate excellent performance of GNN-based decoders along with their low complexity compared to competing methods.
Paper Structure (16 sections, 13 equations, 3 figures, 1 table)

This paper contains 16 sections, 13 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. QLDPC Codes: Design and Decoding
  3. Quantum Stabilizer Codes
  4. Quantum LDPC Codes
  5. Quantum hypergraph (hgp) product code
  6. Quantum bicycle code
  7. Message-Passing Decoding of QLDPC Codes
  8. GNN-Based Decoding of QLDPC Codes
  9. Graph Neural Networks
  10. GNN-based Decoding of QLDPC Codes
  11. Performance Evaluation
  12. Data Set Generation
  13. Quantum Hypergraph--Product (hgp) Code
  14. Quantum Bicycle Code
  15. Comment on Decoder Complexity Scaling
  16. ...and 1 more sections

Figures (3)

  • Figure 1: QLDPC code parity-check matrix $H$ represented as a Tanner graph and its respective underlying GNN structure (with message flow).
  • Figure 2: Performance comparison of the proposed GNN-based QLDPC decoder versus other decoders presented in the literature for different physical error rates $p_f$ (HPG code, $[[n, k]]=[[129, 28]]$). (Note: NBP results are, under identical setting, adopted from Liu.)
  • Figure 3: Performance comparison of the proposed GNN-based QLDPC decoder versus other decoders presented in the literature for different physical error rates $p_f$ (Bicycle code, $[[n, k]]=[[256, 32]]$). (Note: NBP results are, under identical setting, adopted from Liu.)