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Symmetry-tied neural decoders for quantum error correction

An ML decoder tied to the surface code's exact symmetry — at literally identical parameter count — is ~4× more sample-efficient and reaches the matching-decoder baseline where the plain network still lags.

What was measured

Decoding is the practical bottleneck of fault tolerance, and real-device syndrome data is the scarce resource. QEC codes have exact lattice symmetries, so weight-tying should buy sample efficiency — but the naive version fails: a lattice-invariant decoder came out ~10 points worse than plain, because the symmetry maps the logical operator to an equivalent-up-to-stabilizers representative. The correct formulation is a twisted equivariance, (syndromes, label) → (P·syndromes, label ⊕ χ·syndromes), with the twist mask χ solved exactly from the detector error model by GF(2) elimination and verified mechanism-by-mechanism before use.

Headline numbers

• At 250 training samples: plain 0.937 ± 0.007 vs twisted-invariant 0.979 ± 0.004
• ~4× sample efficiency: invariant at 16k ≈ plain at 64k
• At 64k: invariant 0.9943 ± 0.0002 vs MWPM baseline 0.9948 — statistically at the matching decoder; plain trails at 0.9908
• Identical architecture and parameter count (21,377); 5 seeds; 200k test shots; surface code d=3, phenomenological noise p=0.008

Method & discipline

Syndromes sampled with stim; MWPM baseline from pymatching on the exact detector error model. A verifier accepts a symmetry only if it maps the error model onto itself with the observable twist exactly — which surfaced a second finding: circuit-level noise breaks naive spatial symmetry via the CNOT extraction schedule's hook errors. Schedule-aware group actions are the open frontier.

Part of a ~30-experiment program run under one hard rule: no result is recorded before its run completes. Methods and logs: github.com/dwatces.