Train on a small code. Deploy on a large one.
Quantum error correction for topological codes (arXiv:2603.21730): belief propagation decoders fail on these codes because the graph structure contains short cycles that trap the message-passing algorithm. The fix — combining belief propagation with matching-based decoders — works but is computationally expensive. The matching step dominates the complexity.
Neural belief-matching: replace the standard belief propagation with a neural variant. The neural network learns to compensate for the short-cycle problem, producing better input to the matching step and reducing the average decoding complexity. The matching still happens, but on better-prepared inputs, so it converges faster.
The key insight is in training: the toric code has a homogeneous structure — every stabilizer looks the same. A convolutional architecture exploits this homogeneity through weight sharing. The same filters apply everywhere because the code structure is everywhere the same. The consequence: a model trained on a small toric code (few qubits) transfers directly to a larger one (many qubits) without retraining. The learned weights don't depend on code size because the local structure they process doesn't change with code size.
The structural principle: scaling quantum error correction requires decoders that scale with the code. Neural decoders trained on small instances and deployed on large ones solve the training cost problem — you don't need to generate training data at the scale you deploy. The homogeneity of topological codes is what makes this transfer possible. For codes without homogeneity, the trick wouldn't work. The code's regularity is a computational resource for the decoder, not just a mathematical convenience.