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

TL;DR

This work introduces QuBA, a Bayesian graph neural decoder with edge-aware attention for uncertainty-aware decoding of quantum LDPC codes, and SAGU, a sequential training framework to improve cross-domain generalization across code families. By leveraging Monte Carlo predictions and a variational Bayesian formulation, QuBA provides calibrated uncertainty estimates while achieving strong logical error rate reductions on bivariate bicycle codes and their coprime variants. SAGU extends this with a three-phase training protocol that combines diversified domains and parameter averaging to enhance transfer to unseen codes, yielding competitive or superior performance across in-domain and out-of-domain benchmarks. The results show substantial improvements over classical belief propagation and state-of-the-art neural decoders, with up to two orders of magnitude gains in LER under certain conditions, highlighting the practical impact of uncertainty-aware, generalizable decoding for scalable quantum fault tolerance. The work also discusses runtime overhead and broader circuit-level noise as areas for future refinement.

Abstract

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.

Figures (4)

• Figure 1: Relationship between belief propagation (BP) and message passing in graph neural networks (GNNs). (a). A parity-check matrix. (b). The Tanner graph constructed from the parity-check matrix, showing the connections between check nodes and variable nodes (physical qubits). BP operates by exchanging information between check and variable nodes (from check nodes to variable nodes, and vice versa) over $T$ iterations. (c). A message-passing neural network (MPNN) on the Tanner graph. At each iteration, message passing proceeds in two steps. In the aggregation step, every node $v$ computes messages for its neighbors $u \in \mathcal{N}(v)$ by applying a learnable message function $M(\cdot)$. All incoming messages are aggregated at the receiving node using a permutation-invariant operator such as element-wise summation. In the update step, the hidden state of node $v$ is updated by an update function $U(\cdot)$ that combines the previous state with the aggregated messages. After $T$ such iterations, the hidden representation of each node reflects information from its $T$-hop neighborhood.
• Figure 2: Logical error rate (LER) vs. physical error rate ($p$) for BB codes. Lines become vertical when all errors are corrected (LER=0).
• Figure 3: Performance comparison of SAGU and other decoding methods across all domains, with and without OSD, on BB codes. (a): starting domain $[[72, 12, 6]]$; (b): diversity domain $[[144, 12, 12]]$; (c): aggregation domain $[[288, 12, 18]]$; and (d): out-of-domain $[[756, 16, \leq34]]$.
• Figure 4: Logical error rate (LER) vs. physical error rate ($p$) for coprime BB codes.