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Informed Dynamic Scheduling for QLDPC Codes

Tzu-Hsuan Huang, Yeong-Luh Ueng

TL;DR

This work tackles decoding of quantum LDPC codes, where trapping sets and symmetry yield error floors for syndrome-based BP with identical priors. It introduces edge-pool–driven informed dynamic scheduling and a novel predict-and-reduce-error PRE-sRBP, which uses syndrome-based error prediction and parallel trial decoding to diversify updates and mitigate trapping sets. Empirical results on bicycle codes and hypergraph-product codes show PRE-sRBP achieves substantial improvements in frame-error rate with competitive or lower complexity, including over an order of magnitude gains on challenging HP codes. The approach holds promise for low-latency hardware implementations in near-term quantum devices by enabling fast yet robust decoding with parallelizable trials and controlled complexity.

Abstract

Recent research has shown that syndrome-based belief propagation using layered scheduling (sLBP) can not only accelerate the convergence rate but also improve the error rate performance by breaking the quantum trapping sets for quantum low-density parity-check (QLDPC) codes, showcasing a result distinct from classical error correction codes. In this paper, we consider edge-wise informed dynamic scheduling (IDS) for QLDPC codes based on syndrome-based residual belief propagation (sRBP). However, the construction of QLDPC codes and the identical prior intrinsic information assignment will result in an equal residual in many edges, causing a performance limitation for sRBP. Two heuristic strategies, including edge pool design and error pre-correction, are introduced to tackle this obstacle and quantum trapping sets. Then, a novel sRBP equipped with a predict-and-reduce-error mechanism (PRE-sRBP) is proposed, which can provide over one order of performance gain on the considered bicycle codes and symmetric hypergraph (HP) code under similar iterations compared to sLBP.

Informed Dynamic Scheduling for QLDPC Codes

TL;DR

This work tackles decoding of quantum LDPC codes, where trapping sets and symmetry yield error floors for syndrome-based BP with identical priors. It introduces edge-pool–driven informed dynamic scheduling and a novel predict-and-reduce-error PRE-sRBP, which uses syndrome-based error prediction and parallel trial decoding to diversify updates and mitigate trapping sets. Empirical results on bicycle codes and hypergraph-product codes show PRE-sRBP achieves substantial improvements in frame-error rate with competitive or lower complexity, including over an order of magnitude gains on challenging HP codes. The approach holds promise for low-latency hardware implementations in near-term quantum devices by enabling fast yet robust decoding with parallelizable trials and controlled complexity.

Abstract

Recent research has shown that syndrome-based belief propagation using layered scheduling (sLBP) can not only accelerate the convergence rate but also improve the error rate performance by breaking the quantum trapping sets for quantum low-density parity-check (QLDPC) codes, showcasing a result distinct from classical error correction codes. In this paper, we consider edge-wise informed dynamic scheduling (IDS) for QLDPC codes based on syndrome-based residual belief propagation (sRBP). However, the construction of QLDPC codes and the identical prior intrinsic information assignment will result in an equal residual in many edges, causing a performance limitation for sRBP. Two heuristic strategies, including edge pool design and error pre-correction, are introduced to tackle this obstacle and quantum trapping sets. Then, a novel sRBP equipped with a predict-and-reduce-error mechanism (PRE-sRBP) is proposed, which can provide over one order of performance gain on the considered bicycle codes and symmetric hypergraph (HP) code under similar iterations compared to sLBP.
Paper Structure (25 sections, 1 theorem, 11 equations, 14 figures, 1 table, 5 algorithms)

This paper contains 25 sections, 1 theorem, 11 equations, 14 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum Error Codes
  4. Quantum Channel and Syndrome-based BP (sBP)
  5. Quantum Trapping Sets
  6. Informed Dynamic Scheduling (IDS) for QLDPC codes
  7. Syndrome-based IDS
  8. The Edge Pool for sRBP
  9. Error Weight Analysis
  10. Challenges for sRBP on QLDPC codes
  11. Greedy Group Effect on sRBP
  12. Quantum Error Correction Constraint
  13. The Proposed sRBP for QLDPC codes
  14. A Proposed Edge Pool $\mathfrak{R}$ for QLDPC Codes
  15. Error Prediction from the Syndrome
  16. ...and 10 more sections

Key Result

Proposition 1

Considering a QLDPC code where $H_Z=H$ in which the check node degree is a constant $d_c$ over the bit-flip channel with crossover probability $p_x = p$, a priority of the estimated error position belonging to the $\text{supp}(\mathbf{e})$ given the syndrome $\mathbf{s}$ can be determined by the val

Figures (14)

  • Figure 1: The solvable ratio of different error weights using sBP, sLBP, sRBP, and LMD-sRBP where $I_{\max}=90$.
  • Figure 2: Greedy update example. Circles and squares represent variables and check nodes, respectively.
  • Figure 3: The solvable ratio for different error weights by using PRE-sRBP where $\lambda_{\max}=15$ and $I_t=6$.
  • Figure 4: A trapping set example. Those colored in black and red indicate a variable node (check node) that belongs to and does not belong to the support $\text{supp}(\mathbf{e})$ ($\text{supp}(\mathbf{s})$). Edges that are not involved in this set are omitted.
  • Figure 5: The solvable ratio for different error weights using sBP, LMD-sRBP where $I_{\max}=90$, and PRE-sRBP where $\lambda_{\max}=15$, $I_t=6$.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Example 1
  • Example 2