Neural decoders for the surface code
Weights from a study of when learned decoders beat minimum-weight perfect matching, and why. Anyone can train a network to decode a surface code; that has been done since 2017. The question here is narrower. Matching is near-optimal when its noise model is right, so the only place a learned decoder can win is where that model is wrong. This measures how much it wins by, and at what point it stops.
Three architectures, all within 1.2x of each other on parameter count and given identical optimiser, schedule and step budget. Comparing a big model to a small one, or one tuned harder than another, measures capacity or patience rather than architecture.
Using a checkpoint
Weights, the decision threshold picked on validation, and the padding mask all travel in the same file. A checkpoint that makes you reconstruct the threshold by hand is not really reusable.
import torch
ckpt = torch.load("mlp_A_d3_r3_p0.003.pt", weights_only=False)
print(ckpt["architecture"], ckpt["distance"], ckpt["p"], ckpt["logical_error_rate"])
Model definitions live in github.com/Bauxitiego/qec-neural-decoder. Filenames encode architecture, regime, distance, rounds and physical error rate.
The result
Regime B holds total noise constant and varies only its structure: the base error rate is scaled down by bisection until detection-event density matches the uniform control, so a "correlated noise" arm cannot secretly be a "more noise" arm. Matching is run twice, once with the true noise model and once with the uniform one it would actually have on hardware.
| d | noise | MWPM true | MWPM mis-spec | penalty | best neural | vs mis-spec |
|---|---|---|---|---|---|---|
| 3 | s0_x0 |
0.00652 | 0.00652 | 1.00x | 0.00617 | indistinguishable |
| 3 | s0_x0.0005 |
0.01531 | 0.02835 | 1.85x | 0.01536 | neural |
| 3 | s1.75_x0 |
0.00639 | 0.00656 | 1.03x | 0.00591 | neural |
| 3 | s1.75_x0.0015 |
0.01974 | 0.03949 | 2.00x | 0.02006 | neural |
| 3 | s1_x0 |
0.00695 | 0.00702 | 1.01x | 0.00646 | neural |
| 3 | s1_x0.0005 |
0.01229 | 0.02002 | 1.63x | 0.01195 | neural |
| 5 | s0_x0 |
0.00334 | 0.00334 | 1.00x | 0.01111 | mwpm |
| 5 | s0_x0.0005 |
0.01603 | 0.02194 | 1.37x | 0.03611 | mwpm |
| 5 | s1.75_x0 |
0.00290 | 0.00335 | 1.16x | 0.01052 | mwpm |
| 5 | s1.75_x0.0015 |
0.02515 | 0.03575 | 1.42x | 0.04158 | mwpm |
| 5 | s1_x0 |
0.00317 | 0.00356 | 1.12x | 0.01507 | mwpm |
| 5 | s1_x0.0005 |
0.01117 | 0.01439 | 1.29x | 0.02854 | mwpm |
At distance 3 the networks track true-model matching and beat the mis-specified version by up to 2.00x. Two rows are worth reading closely. Per-qubit rate spread on its own barely costs matching anything, around 1.01x, because a mis-weighted graph is still roughly the right graph. Crosstalk costs it 1.85x, because correlated errors have no edge to live on. Rate variation is survivable; correlation is not.
At distance 5 matching wins every row. The next section is why.
Why distance 5 loses
| training shots | epochs | logical error rate | vs MWPM |
|---|---|---|---|
| 120,000 | 8 | 0.04079 | 12.1x |
| 400,000 | 25 | 0.01814 | 5.4x |
| 800,000 | 40 | 0.01118 | 3.3x |
The sweep above ran at 400,000 shots and 12 epochs, between the first two rows. At distance 3 the networks saturate by the second row and settle slightly under matching. At distance 5 they are still improving at the largest budget tested, going 12.1x to 5.4x to 3.3x without flattening.
So the distance-5 numbers are a statement about how long I trained. Read them as an architecture result and you will draw the wrong conclusion, which is the whole reason this section is here.
Regime A: uniform noise
The control. Matching's model is exactly right here, so it should win, and mostly it does.
| d | p | trivial | MWPM | best neural | arch | winner |
|---|---|---|---|---|---|---|
| 3 | 0.001 | 0.02308 | 0.00072 | 0.00092 | mlp | indistinguishable |
| 3 | 0.003 | 0.06567 | 0.00641 | 0.00617 | mlp | indistinguishable |
| 3 | 0.005 | 0.10454 | 0.01668 | 0.01609 | mlp | indistinguishable |
| 3 | 0.01 | 0.18744 | 0.06056 | 0.05562 | mlp | neural |
| 5 | 0.001 | 0.05801 | 0.00010 | 0.00262 | gnn | mwpm |
| 5 | 0.003 | 0.15393 | 0.00338 | 0.01111 | gnn | mwpm |
| 5 | 0.005 | 0.22917 | 0.01401 | 0.03736 | gnn | mwpm |
| 5 | 0.01 | 0.35625 | 0.08121 | 0.15477 | gnn | mwpm |
One row is not a tie: at distance 3 and p=0.01 the MLP beats matching outright with non-overlapping intervals. That was not the expected outcome in the regime designed to favour matching.
The architecture ordering also flips with distance. The plain MLP is best at distance 3, the graph network at distance 5. The graph prior costs more than it returns until the code is large enough for the structure to carry information.
Real device: Google Sycamore
Syndromes from Google's 2023 experiment, with their own published decoder predictions as baselines, scored on identical held-out shots. No reimplementation sits between these models and the comparison.
| d | rounds | trivial | pymatching | correlated | best neural | vs pymatching |
|---|---|---|---|---|---|---|
| 3 | 5 | 0.29340 | 0.14790 | 0.13760 | 0.15720 | indistinguishable |
| 3 | 25 | 0.49010 | 0.43020 | 0.42410 | 0.48920 | mwpm |
| 5 | 5 | 0.38690 | 0.14730 | 0.12750 | 0.26430 | mwpm |
These lose, and the 25-round row loses badly enough to be worth stating plainly: 0.489 against a trivial floor of 0.490 means the model learned essentially nothing. Each experiment has 50,000 shots, so 35,000 to train on, against a 200-detector input with a base flip rate near 50 percent. That is not enough supervision, and no amount of architecture fixes it.
It is also the most practically useful thing in this repo. Simulated syndromes are free. Hardware shots are not, and the shot budget is what actually limits learned decoders on real machines.
Scope
Distance 3 and 5, one code, one crosstalk model. The architectures are standard; the measurement is the part that is mine. State of the art is AlphaQubit from Google DeepMind, which used far more compute and real device data, and nothing here competes with it. Regimes A and B are simulated, so leakage and calibration drift are absent. The distance-5 models remain budget-limited.
License
MIT for code and weights. The device evaluation uses data released by Google Quantum AI under CC-BY-4.0; attribution is on the dataset card.