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Absorbing Sets in Quantum LDPC Codes

Kirsten D. Morris, Tefjol Pllaha, Christine A. Kelley

TL;DR

The paper investigates decoder failures of quantum LDPC codes under syndrome-based iterative decoding, arguing that absorbing sets provide a unifying framework for understanding trapping-set and degenerate-error phenomena in the quantum setting, including $(a,b)$-absorbing sets and their embeddings. It extends the classical absorbing-set framework beyond symmetric stabilizers to a broader, partition-based view, and demonstrates how various graph topologies (e.g., theta graphs, dumbbells, and hypergraph-product codes) yield failure-inducing structures. The results show that most absorbing-set graphs act as trapping sets or failure-inducing configurations under the Gallager-B decoder, and they establish general conditions under which absorbing-set partitions cause decoding failures. These insights guide design principles to mitigate harmful subgraphs in QLDPC codes, with implications for scalable fault-tolerant quantum computation.

Abstract

Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code's graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. The purpose of this paper is to examine failure inducing sets of QLDPC codes under syndrome-based iterative decoding. As for classical LDPC codes, we show that absorbing sets play a central role in understanding decoder failures. Raveendran and Vasic initiated the study of quantum trapping sets, where beyond the classical-type trapping sets, they identified rigid symmetric structures (a.k.a symmetric stabilizers) responsible for degenerate errors. In this paper, we show that this behavior is part of a much more general phenomenon that can be described by the absorbing set framework.

Absorbing Sets in Quantum LDPC Codes

TL;DR

The paper investigates decoder failures of quantum LDPC codes under syndrome-based iterative decoding, arguing that absorbing sets provide a unifying framework for understanding trapping-set and degenerate-error phenomena in the quantum setting, including -absorbing sets and their embeddings. It extends the classical absorbing-set framework beyond symmetric stabilizers to a broader, partition-based view, and demonstrates how various graph topologies (e.g., theta graphs, dumbbells, and hypergraph-product codes) yield failure-inducing structures. The results show that most absorbing-set graphs act as trapping sets or failure-inducing configurations under the Gallager-B decoder, and they establish general conditions under which absorbing-set partitions cause decoding failures. These insights guide design principles to mitigate harmful subgraphs in QLDPC codes, with implications for scalable fault-tolerant quantum computation.

Abstract

Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code's graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. The purpose of this paper is to examine failure inducing sets of QLDPC codes under syndrome-based iterative decoding. As for classical LDPC codes, we show that absorbing sets play a central role in understanding decoder failures. Raveendran and Vasic initiated the study of quantum trapping sets, where beyond the classical-type trapping sets, they identified rigid symmetric structures (a.k.a symmetric stabilizers) responsible for degenerate errors. In this paper, we show that this behavior is part of a much more general phenomenon that can be described by the absorbing set framework.
Paper Structure (11 sections, 15 theorems, 11 equations, 11 figures, 1 table)

This paper contains 11 sections, 15 theorems, 11 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stabilizer formalism
  4. QLDPC codes and syndrome-based iterative decoding
  5. Iterative decoder failure and trapping sets
  6. Absorbing Sets on Their Own
  7. Beyond Symmetric Stabilizers
  8. Symmetric Stabilizers
  9. Extending the absorbing set partition
  10. Hypergraph-product Codes
  11. Concluding Observations

Key Result

Theorem 4

Let ${\mathcal{A}}$ be an $(a,b)$-absorbing set with $b\geq1$. Then ${\mathcal{A}}$ is a failure inducing set. In particular, the decoding syndrome will always be ${0}$ and thus the syndrome value at the odd degree check nodes will never match the input syndrome, thus resulting in a decoding failure

Figures (11)

  • Figure 1: Left: A graph induced by a $(4,2)$-trapping set $\mathcal{T}$. Right: Failure inducing sets of $\mathcal{T}$.
  • Figure 2: Failure inducing sets of a $(5,3)$-trapping set.
  • Figure 3: $(5,5)$-trapping set
  • Figure 4: Non isomorphic $T(6,6,4)$ theta graphs
  • Figure 5: Two iterations of the syndrome-based Gallager-B decoder on an acyclic $(a,0)$-absorbing set. The input syndrome is depicted in red, the outgoing variable node messages in blue, and the outgoing check node messages in cyan.
  • ...and 6 more figures

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 1
  • Definition 5
  • Example 2
  • Example 3
  • Theorem 4
  • proof
  • ...and 37 more