Degenerate quantum erasure decoding

TL;DR

This work investigates erasure-dominated quantum error correction using stabilizer and QLDPC codes, targeting erasure capacity under practical, linear-time decoding. The authors show that MLD can achieve capacity on the quantum erasure channel with $C(p)=1-2p$, while novel BP-based decoders—MBP$_2$, AMBP$_2$, MBP$_4$, and the unified (A)MBP$_q$—achieve near-capacity across diverse code families, including bicycle codes, LP codes, and 2D topological codes, with linear-time complexity. A key strength is exploiting degeneracy in stabilizer codes to converge to any degenerate equivalent error; the decoders also extend to mixed erasure/depoloarizing errors and local deletions via PI inner codes. The results demonstrate capacity-achieving performance for certain constructions (MLD) and near-capacity performance for linear-time BP decoders, suggesting practical pathways to high-rate quantum codes with efficient decoding for leakage-dominated quantum devices and networks.

Abstract

Erasures are the primary type of errors in physical systems dominated by leakage errors. While quantum error correction (QEC) using stabilizer codes can combat erasure errors, it remains unknown which constructions achieve capacity performance. If such codes exist, decoders with linear runtime in the code length are also desired. In this paper, we present erasure capacity-achieving quantum codes under maximum-likelihood decoding (MLD), though MLD requires cubic runtime in the code length. For QEC, using an accurate decoder with the shortest possible runtime will minimize the degradation of quantum information while awaiting the decoder's decision. To address this, we propose belief propagation (BP) decoders that run in linear time and exploit error degeneracy in stabilizer codes, achieving capacity or near-capacity performance for a broad class of codes, including bicycle codes, product codes, and topological codes. We furthermore explore the potential of our BP decoders to handle mixed erasure and depolarizing errors, and also local deletion errors via concatenation with permutation invariant codes.

Figures (29)

• Figure 1: Comparison of decoders. The vertical axis indicates complexity, while the horizontal axis shows code types with increasing connectivity from left to right. BPs in this paper perform accurately across various code types, similar to the Gaussian decoder, with a slight threshold gap but with linear complexity. The achieved thresholds are shown in Fig. \ref{['fig:bnd_logx']}.
• Figure 2: Quantum erasure capacity and thresholds of codes. For 2D topological codes, BP and MLD share a threshold of $p_\text{th} = 0.5$. For other codes, a small accuracy gap is indicated by two points: $p^\text{(BP)}_\text{th}$ (left) and $p^\text{(MLD)}_\text{th}$ (right). The constructed codes achieve capacity with MLD, $r = 1 - 2 p^\text{(MLD)}_\text{th}$, and near-capacity with BP, $r = 1 - 2.5 p^\text{(BP)}_\text{th}$. Higher-rate codes require larger stabilizer weight $w$, consistent with the results in DZ13 ([DZ13]).
• Figure 3: (a) Tanner graph induced by the binary matrix $\bm{H}$ in \ref{['eq:4_1_code_H2']}. (b) Tanner graph induced by the Pauli matrix $H$ in \ref{['eq:4_1_code_H4']}, with edge types $X$ (solid line), $Y$ (dashed line), and $Z$ (dotted line).
• Figure 4: Local deletion to erasure rate conversion using $[[\ell,1]]$ PI codes for $\ell = 4,7,9,15$ (Yin14HagiwaraISIT2020, PoR04, Rus00, AAB24). The $[[4,1]]$ PI code converts $\epsilon=0.385$ to $p=0.5$, which can then be handled by topological codes. At $\epsilon=0.1$, an $[[\ell<15,\,1]]$ PI code achieves $p<0.1$, allowing a QLDPC code rate above 3/4 (Fig. \ref{['fig:bnd_logx']}). At $\epsilon=0.01$, using a 7-qubit PI code achieves $p<10^{-4}$, which allows QLDPC codes to attain a rate of essentially 1.
• Figure 5: BP accuracy for mixed erasure and depolarizing errors using the $[[1054,140]]$ LP code. The curves with only erasure errors are provided for reference ($p_\text{dep}=0$). AMBP$_4$ maintains strong accuracy in the mixed error model ($p_\text{dep}=0.01$).

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Lemma 1
• Theorem 2
• Example 1
• Remark 3
• Definition 3
• Definition 4
• Definition 5
• Lemma 4