On the Efficacy of the Peeling Decoder for the Quantum Expander Code

TL;DR

This work tackles erasure decoding for quantum LDPC codes, focusing on quantum expander codes produced by hypergraph products. It introduces a two-phase linear-time erasure decoder: first applying a classical peeling step on the $Z$-type Tanner graph, then clustering the remaining erasures into vertical and horizontal components and solving isolated/frozen clusters with Viderman-based decoding, with optional small-set-flip steps for residual cycles. A detailed analysis (Sections 5.1–5.3) establishes when peeling and SSF succeed under expansion assumptions and characterizes the algorithmic complexity; simulations on several large quantum expander codes show that longer codes improve failure rates at fixed erasure rates with linear-time peeling, while residual errors can be mitigated by subsequent processing. Overall, the results demonstrate a practical, scalable erasure-decoding strategy for quantum expander codes, offering a viable path toward efficient quantum LDPC decoders and motivating further refinements to address remaining stopping-set patterns and residual errors.

Abstract

The problem of recovering from qubit erasures has recently gained attention as erasures occur in many physical systems such as photonic systems, trapped ions, superconducting qubits and circuit quantum electrodynamics. While several linear-time decoders for error correction are known, their error-correcting capability is limited to half the minimum distance of the code, whereas erasure correction allows one to go beyond this limit. As in the classical case, stopping sets pose a major challenge in designing efficient erasure decoders for quantum LDPC codes. In this paper, we show through simulation, that an attractive alternative here, is the use of quantum expander codes in conjunction with the peeling decoder that has linear complexity. We also discuss additional techniques including small-set-flip decoding, that can be applied following the peeling operation, to improve decoding performance and their associated complexity.

Figures (5)

• Figure 1: The Tanner graph for the HGP code constructed from classical codes $\mathcal{C}_1$ with Tanner graph $\mathcal{T}(V_1 \cup C_1, E_1)$ and $\mathcal{C}_2$ with Tanner graph $\mathcal{T}(V_2 \cup C_2, E_2$).
• Figure 2: Stabilizer stopping set for the HGP code constructed from two classical $3$-bit repetition code.
• Figure 3: Horizontal and vertical stopping sets for the HGP code.
• Figure 4: Illustration of different types of clusters.
• Figure 5: Performance of the $[[1525,25]]$, $[[3904,64]]$, $[[6100,100]]$, and $[[8784,144]]$ quantum expander codes under the peeling decoder.

Theorems & Definitions (11)

• Definition 1
• Theorem 1
• Definition 2
• Remark 1
• Theorem 2
• proof
• Definition 3
• Lemma 1
• Lemma 2
• Theorem 3