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Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided Decimation

Mert Gökduman, Hanwen Yao, Henry D. Pfister

TL;DR

This work shows that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes and demonstrates BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape.

Abstract

Quantum low-density parity-check (LDPC) codes are a promising family of quantum error-correcting codes for fault tolerant quantum computing with low overhead. Decoding quantum LDPC codes on quantum erasure channels has received more attention recently due to advances in erasure conversion for various types of qubits including neutral atoms, trapped ions, and superconducting qubits. Belief propagation with guided decimation (BPGD) decoding of quantum LDPC codes has demonstrated good performance in bit-flip and depolarizing noise. In this work, we apply BPGD decoding to quantum erasure channels. Using a natural modification, we show that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes. Furthermore, we show that the performance of BPGD decoding on erasure channels can sometimes be improved significantly by either adding damping or adjusting the initial channel log-likelihood ratio for bits that are not erased. More generally, our results demonstrate BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape.

Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided Decimation

TL;DR

This work shows that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes and demonstrates BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape.

Abstract

Quantum low-density parity-check (LDPC) codes are a promising family of quantum error-correcting codes for fault tolerant quantum computing with low overhead. Decoding quantum LDPC codes on quantum erasure channels has received more attention recently due to advances in erasure conversion for various types of qubits including neutral atoms, trapped ions, and superconducting qubits. Belief propagation with guided decimation (BPGD) decoding of quantum LDPC codes has demonstrated good performance in bit-flip and depolarizing noise. In this work, we apply BPGD decoding to quantum erasure channels. Using a natural modification, we show that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes. Furthermore, we show that the performance of BPGD decoding on erasure channels can sometimes be improved significantly by either adding damping or adjusting the initial channel log-likelihood ratio for bits that are not erased. More generally, our results demonstrate BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape.

Paper Structure

This paper contains 17 sections, 19 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical Erasure Correction
  4. Stabilizer Formalism
  5. Syndrome Decoding for the Quantum Erasure Channel
  6. Quantum LDPC Codes
  7. Decoding Erasures with BPGD
  8. Belief Propagation Decoding
  9. BP Convergence Issues for Quantum LDPC Codes
  10. Belief Propagation with Guided Decimation
  11. BPGD Decoding for the Quantum Erasure Channel
  12. BPGD with Damping and Adjusted LLRs
  13. BPGD with Adjusted LLRs
  14. BPGD with Damping
  15. BPGD with Combined Adjustments
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Comparison the quantum erasure channel of BPGD decoding with peeling, pruned peeling, VH, and ML decoding. The plot presents the decoder failure rates for the HGP QLDPC code [[1600, 64]] from connolly2024fast, with convergence to a degenerate codeword considered a success. The number of simulations per data point varies and was chosen to ensure short error bars.
  • Figure 2: Comparison on the quantum erasure channel of BPGD decoding with peeling, pruned peeling, VH, and ML decoding. The plot presents the decoder failure rates for the HGP QLDPC code [[2025, 81]] from connolly2024fast.
  • Figure 3: Comparison of BPGD decoding over the quantum erasure channel with peeling, pruned peeling, and ML decoding. The curves show the decoder failure rates for the [[882, 24, $18 \leqslant d \leqslant 24$]] B1 lifted-product QLDPC code from Panteleev2021degeneratequantum.