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Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation

Hanwen Yao, Waleed Abu Laban, Christian Häger, Alexandre Graell i Amat, Henry D. Pfister

TL;DR

This work tackles convergence challenges in belief propagation (BP) decoding for degenerate quantum LDPC codes by introducing BP guided decimation (BPGD). By combining BP with a decimation-based mechanism that sequentially fixes the most reliable qubits, BPGD imitates sampling-based decoding without solving linear systems and achieves performance on par with BP-OSD and BP-SI. The authors also extend the approach to quaternary BP for depolarizing noise and conduct randomized experiments to illuminate the role of code degeneracy in decoding. Overall, BPGD offers a low-complexity, scalable alternative that improves BP convergence and leverages degeneracy to enhance decoding success in quantum LDPC codes.

Abstract

Quantum low-density parity-check (QLDPC) codes have emerged as a promising technique for quantum error correction. A variety of decoders have been proposed for QLDPC codes and many of them utilize belief propagation (BP) decoding in some fashion. However, the use of BP decoding for degenerate QLDPC codes is known to have issues with convergence. These issues are typically attributed to short cycles in the Tanner graph and code degeneracy (i.e. multiple error patterns with the same syndrome). Although various methods have been proposed to mitigate the non-convergence issue, such as BP with ordered statistics decoding (BP-OSD) and BP with stabilizer inactivation (BP-SI), achieving better performance with lower complexity remains an active area of research. In this work, we propose a decoder for QLDPC codes based on BP guided decimation (BPGD), which has been previously studied for constraint satisfaction and lossy compression problems. The decimation process is applicable to both binary and quaternary BP and it involves sequentially fixing the value of the most reliable qubits to encourage BP convergence. Despite its simplicity, We find that BPGD significantly reduces the BP failure rate due to non-convergence, achieving performance on par with BP with ordered statistics decoding and BP with stabilizer inactivation, without the need to solve systems of linear equations.

Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation

TL;DR

This work tackles convergence challenges in belief propagation (BP) decoding for degenerate quantum LDPC codes by introducing BP guided decimation (BPGD). By combining BP with a decimation-based mechanism that sequentially fixes the most reliable qubits, BPGD imitates sampling-based decoding without solving linear systems and achieves performance on par with BP-OSD and BP-SI. The authors also extend the approach to quaternary BP for depolarizing noise and conduct randomized experiments to illuminate the role of code degeneracy in decoding. Overall, BPGD offers a low-complexity, scalable alternative that improves BP convergence and leverages degeneracy to enhance decoding success in quantum LDPC codes.

Abstract

Quantum low-density parity-check (QLDPC) codes have emerged as a promising technique for quantum error correction. A variety of decoders have been proposed for QLDPC codes and many of them utilize belief propagation (BP) decoding in some fashion. However, the use of BP decoding for degenerate QLDPC codes is known to have issues with convergence. These issues are typically attributed to short cycles in the Tanner graph and code degeneracy (i.e. multiple error patterns with the same syndrome). Although various methods have been proposed to mitigate the non-convergence issue, such as BP with ordered statistics decoding (BP-OSD) and BP with stabilizer inactivation (BP-SI), achieving better performance with lower complexity remains an active area of research. In this work, we propose a decoder for QLDPC codes based on BP guided decimation (BPGD), which has been previously studied for constraint satisfaction and lossy compression problems. The decimation process is applicable to both binary and quaternary BP and it involves sequentially fixing the value of the most reliable qubits to encourage BP convergence. Despite its simplicity, We find that BPGD significantly reduces the BP failure rate due to non-convergence, achieving performance on par with BP with ordered statistics decoding and BP with stabilizer inactivation, without the need to solve systems of linear equations.
Paper Structure (23 sections, 1 theorem, 45 equations, 8 figures, 2 tables, 3 algorithms)

This paper contains 23 sections, 1 theorem, 45 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Binary Linear Codes
  4. Stabilizer Formalism
  5. Syndrome Decoding of Stabilizer Codes
  6. Quantum Maximum Likelihood (QML) Decoding
  7. Degenerate Quantum Maximum Likelihood (DQML) Decoding
  8. CSS Codes with Bit-Flip Errors
  9. Belief Propagation Decoding
  10. Belief Propagation for Bit-Flip Errors
  11. Prior Improvements on BP
  12. Belief Propagation with Guided Decimation
  13. Sampling Decoding for Stabilizer Codes
  14. Approximating the Sampling Decoder
  15. The BPGD Algorithm
  16. ...and 8 more sections

Key Result

Theorem 1

Consider decoding a stabilizer code over Pauli errors following some distribution. Denote the error probability of the DQML decoder by $P_{\mathrm{DQML}}$, and denote the error probability of the sampling decoder by $P_{\mathrm{S}}$. Then, the following inequalities hold:

Figures (8)

  • Figure 1: Tanner graph of $H_1$ for the $[[7,1,3]]$ Steane code
  • Figure 2: Performance of BPGD on the $[[882,24,18\leqslant d\leqslant 24]]$ B1 code Panteleev2021degeneratequantum over bit-flip noise following different decimation orders. The data points are collected by running simulations until we observe 100 error cases.
  • Figure 3: Performance of BPGD on the $[[882,24,18\leqslant d\leqslant 24]]$ B1 code Panteleev2021degeneratequantum over bit-flip noise with different BP iterations per round. The data points are collected by running simulations until we observe 100 error cases.
  • Figure 4: Performance of various decoders on the $[[882,24,18\leqslant d\leqslant 24]]$ B1 code Panteleev2021degeneratequantum over bit-flip noise. The data points of BP, BP-OSD with order 0, and BPGD decoders are collected by running simulations until we observe 100 error cases. The data points of BP-SI are taken and translated from du2022stabilizer
  • Figure 5: Performance of various decoders on the $[[1922,50,16]]$ C2 code Panteleev2021degeneratequantum over bit-flip noise. The data points of BP, BP-OSD with order 0, and BPGD decoders are collected by running simulations until we observe 100 error cases. The data points of BP-SI are taken and translated from du2022stabilizer
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1: DQML vs. sampling