Efficient and Universal Neural-Network Decoder for Stabilizer-Based Quantum Error Correction

TL;DR

The paper introduces GraphQEC, a universal neural-network decoder for stabilizer-based quantum error correction that operates directly on the graph structure of codes. By formulating decoding as a temporal graph-classification problem on an extended Tanner graph and employing a three-phase architecture (encoder, decoder, readout) with linear-time components, GraphQEC achieves high accuracy across color codes, BB codes, and surface codes while maintaining real-time decoding speeds. Key contributions include a topology-agnostic graph neural network that outperforms specialized decoders across multiple code families, and demonstration of improved decoding thresholds and break-even performance via sub-threshold scaling analysis. The work demonstrates the practicality of a single, scalable neural decoder for diverse stabilizer codes, paving the way for real-time fault-tolerant quantum computing at scale.

Abstract

Scaling quantum computing to practical applications necessitates reliable quantum error correction. Although numerous correction codes have been proposed, the overall correction efficiency critically limited by the decode algorithms. We introduce GraphQEC, a code-agnostic decoder leveraging machine-learning on the graph structure of stabilizer codes with linear time complexity. GraphQEC demonstrates unprecedented accuracy and efficiency across all tested code families, including surface codes, color codes, and quantum low-density parity-check (QLDPC) codes. For instance, on a distance-12 QLDPC code, GraphQEC achieves a logical error rate of $9.55 \times 10^{-5}$, an 18-fold improvement over the previous best specialized decoder's $1.74 \times 10^{-3}$ under $p=0.005$ physical error rates, while maintaining $157μ$s/cycle decoding speed. Our approach represents the first universal solution for real-time quantum error correction across arbitrary stabilizer codes.

Figures (8)

• Figure 1: Overview of the stabilizer code decoding framework. (A) The stabilizer code can be represented by a stabilizer generator matrix, and the logical qubit is defined by a set of selected single-qubit pauli operators. (B) The stabilizer code can be equivalently represented by an extended Tanner Graph, with physical qubits become data nodes, stabilizer generators become check nodes, and logical pauli operators become logical nodes. (C) The syndrome measurement circuit generate the syndromes repeatedly. The GraphQEC decoder receives the syndromes and decode it on the extended Tanner graph.
• Figure 2: Comparative Performance Analysis of Quantum Error Correction Decoders. (A) GraphQEC logical error rates (LER) on a logarithmic scale (color bar). (B) Baseline decoder (BP-OSD/Concat-Matching) performance under identical conditions. White contours indicate break-even regions where logical error rates fall below physical rates. (C) Performance ratio LERGraphQEC/LERBaseline demonstrates GraphQEC's superior correction capability, with ratios approaching $<1\%$ in low-error regimes.
• Figure 3: Architecture of the GraphQEC decoder. The leftmost flowchart illustrates the high-level structure of our proposed quantum error correction neural network, with colored blocks detailed on the right. White blocks represent operators and gray blocks represent variables. Superscript indices denote time slices, while subscript indices differentiate between node types in the quantum code graph: $i$ for check nodes, $n$ for data nodes, and $k$ for logical nodes. The shadowed linear attention block operates across the temporal dimension, enabling information flow between time slices, while all other operators process spatial relationships within the graph at each individual time step.
• Figure 4: Decoding Time for different Decoders on BB code. Decoding times are plotted for the d=6 and d=12 BB code. We test it under a physical error rate of $p=0.005$ and take the BP-OSD method for reference. The x-axis is broken to show both short and long decoding times. The GraphQEC decoder is much faster than the BP based decoders and keeps linear time scaling up to thousands of cycles.
• Figure S1: The sub-threshold scaling of decoder performance on different code. The data points with different marker represents different decoders. The dash-lines is the fitting curve, where fitting parameters are decided separately for each decoder and each code family, and is shared inside each group. The $d=3$ color code is excluded from the fitting to reduce the influence of finite size effect.