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Paper 1

Improved decoding algorithms for surface codes under independent bit-flip and phase-flip errors

Louay Bazzi

Year: 2026
Journal: arXiv preprint
DOI: arXiv:2601.00972
arXiv: 2601.00972

We study exact decoding for the toric code and for planar and rotated surface codes under the standard independent $X/Z$ noise model, focusing on Separate Minimum Weight (SMW) decoding and Separate Most Likely Coset (SMLC) decoding. For the SMW decoding problem, we show that an $O(n^{3/2}\log n)$-time decoder is achievable for surface and toric codes, improving over the $O(n^{3}\log n)$ worst-case time of the standard approach based on complete decoding graphs. Our approach is based on a local reduction of SMW decoding to the minimum weight perfect matching problem using Fisher gadgets, which preserves planarity for planar and rotated surface codes and genus~$1$ for the toric code. This reduction enables the use of Lipton--Tarjan planar separator methods and implies that SMW decoding lies in $\mathrm{NC}$. For SMLC decoding, we show that the planar surface code admits an exact decoder with $O(n^{3/2})$ algebraic complexity and that the problem lies in $\mathrm{NC}$, improving over the $O(n^{2})$ algebraic complexity of Bravyi \emph{et al.} Our approach proceeds via a dual-cycle formulation of coset probabilities and an explicit reduction to planar Pfaffian evaluation using Fisher--Kasteleyn--Temperley constructions. The same complexity measures apply to SMLC decoding of the rotated surface code. For the toric code, we obtain an exact polynomial-time SMLC decoder with $O(n^{3})$ algebraic complexity. In addition, while the SMLC formulation is motivated by connections to statistical mechanics, we provide a purely algebraic derivation of the underlying duality based on MacWilliams duality and Fourier analysis. Finally, we discuss extensions of the framework to the depolarizing noise model and identify resulting open problems.

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Paper 2

Toward Uncertainty-Aware and Generalizable Neural Decoding for Quantum LDPC Codes

Xiangjun Mi, Frank Mueller

Year: 2025
Journal: arXiv preprint
DOI: arXiv:2510.06257
arXiv: 2510.06257

Quantum error correction (QEC) is essential for scalable quantum computing, yet decoding errors via conventional algorithms result in limited accuracy (i.e., suppression of logical errors) and high overheads, both of which can be alleviated by inference-based decoders. To date, such machine-learning (ML) decoders lack two key properties crucial for practical fault tolerance: reliable uncertainty quantification and robust generalization to previously unseen codes. To address this gap, we propose \textbf{QuBA}, a Bayesian graph neural decoder that integrates attention to both dot-product and multi-head, enabling expressive error-pattern recognition alongside calibrated uncertainty estimates. Building on QuBA, we further develop \textbf{SAGU }\textbf{(Sequential Aggregate Generalization under Uncertainty)}, a multi-code training framework with enhanced cross-domain robustness enabling decoding beyond the training set. Experiments on bivariate bicycle (BB) codes and their coprime variants demonstrate that (i) both QuBA and SAGU consistently outperform the classical baseline belief propagation (BP), achieving a reduction of on average \emph{one order of magnitude} in logical error rate (LER), and up to \emph{two orders of magnitude} under confident-decision bounds on the coprime BB code $[[154, 6, 16]]$; (ii) QuBA also surpasses state-of-the-art neural decoders, providing an advantage of roughly \emph{one order of magnitude} (e.g., for the larger BB code $[[756, 16, \leq34]]$) even when considering conservative (safe) decision bounds; (iii) SAGU achieves decoding performance comparable to or even outperforming QuBA's domain-specific training approach.

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