PaperlayerView exact revision on arXiv

Source-linked AI summary

Learning to Decode the Surface Code with a Recurrent, Transformer-Based Neural Network

Johannes Bausch, Andrew W Senior, Francisco J H Heras, Thomas Edlich, Alex Davies, Michael Newman, Cody Jones, Kevin Satzinger, Murphy Yuezhen Niu, Sam Blackwell, George Holland, Dvir Kafri, Juan Atalaya, Craig Gidney, Demis Hassabis, Sergio Boixo, Hartmut Neven, Pushmeet Kohli

arXiv:2310.05900v1quant-phcs.LG

TL;DR

Reliable quantum computation requires correcting physical errors so deep circuits can run confidently. This paper presents a recurrent neural-network decoder for the surface code that achieves state-of-the-art error suppression, outperforming the previous best tensor-network decoder on experimental data and retaining its advantage at distances up to 11 and beyond its training-cycle regime.

  • Problem

    Practical quantum computation requires correcting physical errors so deep circuits can run with high confidence.

    Source: §1.1 · p. 2

  • Method

    The paper uses a recurrent neural-network decoder whose fixed-size state stores stabilizer observations across cycles.

    Source: §2.1 · p. 4

  • Results

    The decoder achieves state-of-the-art error suppression, outperforms the previous best tensor-network decoder on experimental data, and retains its accuracy advantage up to distance 11 beyond its training-cycle regime.

    Source: §3 · pp. 15–16

  • Takeaways & Limitations

    The results support machine learning as a contender for decoding surface codes across larger distances and longer error-correction sequences.

    Source: §3 · pp. 15–16

    Achieving real-time throughput rates for superconducting architectures remains a challenge.

    Source: §3 · p. 16

Abstract

from arXiv · show

Quantum error-correction is a prerequisite for reliable quantum computation. Towards this goal, we present a recurrent, transformer-based neural network which learns to decode the surface code, the leading quantum error-correction code. Our decoder outperforms state-of-the-art algorithmic decoders on real-world data from Google's Sycamore quantum processor for distance 3 and 5 surface codes. On distances up to 11, the decoder maintains its advantage on simulated data with realistic noise including cross-talk, leakage, and analog readout signals, and sustains its accuracy far beyond the 25 cycles it was trained on. Our work illustrates the ability of machine learning to go beyond human-designed algorithms by learning from data directly, highlighting machine learning as a strong contender for decoding in quantum computers.

1. Quantum error correction

Fault-tolerant quantum computation requires accurate, fast decoding of surface-code syndromes under realistic noise. This work introduces a recurrent transformer-convolutional decoder that learns directly from data and improves accuracy across experimental and simulated settings.

1.1. Context and background

  • The surface code protects logical qubits by redundantly encoding them in physical-qubit grids and detecting errors through repeated stabilizer measurements.Its code distance is the minimum length of logical observables and, for square codes, equals the data-qubit grid side length.Source: §1.1 · p. 2
  • Accurate, fast decoding is essential because realistic quantum noise, degeneracy, leakage, and crosstalk complicate syndrome interpretation and can otherwise create processing backlogs.Decoder performance also affects the required code size or processor gate fidelity for fault-tolerant computation.Source: §1.1 · pp. 2–3

1.2. Quantum error correction with machine learning

  • The work presents a recurrent architecture combining transformers and convolutions to decode the surface code in fault-tolerant circuit-level noise.It learns from raw data and expands machine-learning decoding to larger scale and accuracy than earlier approaches.Source: §1.2 · p. 4
  • The decoder achieves markedly better error suppression than state-of-the-art correlated matching and tensor network decoders on Google Sycamore surface-code data [43] [14].For larger code distances up to 11, it also outperforms state-of-the-art correlated matching decoders under realistic leakage, crosstalk, and analog-readout noise, sustaining accuracy for at least 100 000 cycles despite training on up to 25.Source: §1.2 · p. 4
  • The model incorporates raw in-phase and quadrature readout signals and leakage information while providing calibrated error probabilities without computational overhead.These capabilities arise from learning directly from device data and syndrome information.Source: §1.2 · p. 4

2. A recurrent syndrome transformer

This section presents a recurrent Syndrome Transformer decoder that maintains per-stabilizer state and learns structured information flow across error-correction cycles. It describes the decoder’s accuracy, robustness, calibrated confidence, rich inputs, scalable and hardware-specific operation, and generalization to long experiments under realistic noise conditions.

2.1. Model architecture

  • The recurrent architecture repeatedly updates a fixed-size decoder state that stores stabilizer information through the current cycle.It maintains a vector representation per stabilizer, incorporates each cycle’s measurements and events, and uses a Syndrome Transformer with attention bias and spatial convolutions to model physical relationships.Source: §2.1 · pp. 4–5

2.2. Decoding the Sycamore memory experiment

  • 2.901 ± 0.023% and 2.748 ± 0.015% LER at distances 3 and 5, respectively, beat the tensor network decoder’s 3.028 ± 0.023% and 2.915 ± 0.016%.The complete procedure combines experimental fine-tuning and ensembling, producing a clear advantage over tensor networks and MWPM variants on Sycamore data.Source: §2.2 · pp. 7–8, Figure 2 · p. 8

2.3. Beyond Pauli noise: leakage, cross-talk, I/Q noise

  • To support scaling to higher code distances, the study explores decoder performance using simulated data at error rates comparable to and significantly lower than Sycamore experimental data.The study independently modulates effects at code distances 3, 5, 7, 9, and 11.Source: §2.3 · p. 9

2.3.1. Analog readouts

  • Providing probabilistic I/Q-derived inputs improves performance for both 3 × 3 and 5 × 5 codes, while the ML decoder processes them without runtime penalty.The inputs replace binary stabilizer values with probabilities of the |1⟩ state and probabilistic detection events; MWPM-Corr also improves but incurs a runtime penalty.Source: §2.3.1 · p. 9

2.3.2. Leakage and crosstalk

  • At distance 11, under leakage and crosstalk, the ML decoder achieves 5.37 ± 0.01 × 10−6 LER versus MWPM-Corr’s 6.74±0.02×10−6, outperforming it.The ML-based decoder is more accurate than MWPM-Corr across distances 3–9 when trained on 2 × 10^9 samples; average suppression from distances 3 to 11 is Λ = 4.28 ± 0.02.Source: §2.3.2 · p. 10
  • At code distances 3–9, the ML decoder outperforms MWPM-Corr with binary inputs and achieves parity at distance 11, while leakage and richer inputs further improve its advantage.At distance 7 with 0.1% leakage, the ML decoder’s logical error per round increases by ≈30%, compared with ≈60% −80% for leakage-unaware MWPM-Corr; training on up to 25 cycles generalized to at least 100 000 cycles.Source: §2.3.2 · pp. 10–11, Figure 3 · p. 12

2.4. Postselection: improving accuracy with confidence

  • Postselection lowers the error rate by a factor of ≈10, while 10% rejection lowers it by a factor of ≈250.The decoder’s output probability is well-calibrated: among samples assigned probability 0.8, approximately 80% have the true label |1⟩.Source: §2.4 · p. 13

2.5.1. Advantages

  • The decoder can train directly on raw hardware data, process analog, I/Q, leakage, and other side information, and output calibrated error probabilities for postselection.These capabilities reduce dependence on accurately characterized noise models and improve accuracy over stabilizer-syndrome inputs alone.Source: §2.5.1 · p. 14

2.5.2. Throughput considerations

  • On an accelerator, throughput for code distances up to 25 is within about one to two orders of magnitude of the 1µs target and is independent of noise rate.The recurrent architecture supports indefinite cycles, parallel processing, and efficient scaling across code distance with one fixed-size architecture, despite not yet being optimized for inference speed.Source: §2.5.2 · p. 15

2.5.3. Training data requirements

  • Training-data requirements grow exponentially with code distance, requiring a hundredfold increase from distance 3 to distance 11 and an estimated 10^13 −10^14 examples for distance 25 parity.Generic-noise-model pretraining can be followed by rapid hardware-specific fine-tuning rather than training a new model from scratch.Source: §2.5.3 · p. 15

3. Conclusions and outlook

The neural-network decoder achieves state-of-the-art surface-code error suppression and outperforms the previous best-in-class tensor-network decoder on experimental data, though real-time superconducting throughput remains challenging. Its accuracy advantage persists up to distance 11 and beyond its training-cycle regime, while its ability to exploit rich experimental inputs supports the prospect of practical quantum computing.

Conclusions and outlook

  • The decoder’s architecture incorporates inductive biases motivated by quantum error correction to learn surface-code decoding directly.This learning-based design underlies the reported error-suppression performance.Source: §3 · p. 15
  • The neural-network decoder achieves state-of-the-art surface-code error suppression and outperforms the previous best-in-class tensor-network decoder on experimental data, which takes orders of magnitude longer to run.Achieving real-time throughput rates of a superconducting architecture remains a challenge.Source: §3 · pp. 15–16
  • Its accuracy advantage over correlated matching persists at distances up to 11 and across error-correction cycles far beyond training, but extending suppression beyond distance 11 remains challenging.The architecture can execute at larger code distances with only a moderate runtime increase.Source: §3 · p. 16
  • The model learns from real experimental data and uses rich inputs representing I/Q noise and leakage without requiring a human to design an explicit algorithm for each feature.This ability to use available experimental information showcases machine learning’s strength in a scientific context.Source: §3 · p. 16
  • The results provide evidence that machine-learning decoders may achieve the error suppression and speed needed for practical quantum computing.The authors expect other decoding techniques to continue improving.Source: §3 · p. 16

Authors contributions … A.1.2. The rotated surface code

The paper’s supporting work spans realistic noise modeling, experimental and theoretical decoder expertise, and a rotated-surface-code memory experiment framework. The methods describe repeated stabilizer readouts on a checkerboard data-qubit grid, with code-specific circuits and multiple noise-model abstractions.

Authors contributions

  • J.A. and D.K. developed models and software for modeling realistic noise in superconducting processors.Source: Authors contributions · p. 27
  • J.B., S.B., C.J., C.G., D.H., P.K., and H.N. contributed research conceptualization, supervision, decoder theory, noise-model configuration, software, and scalable-decoding direction.Source: Authors contributions · p. 27
  • Other contributors supported investigation, methodology, data curation, validation, visualization, experimental leakage and soft-information analysis, project administration, and paper writing.Source: Authors contributions · p. 27

A.1.1. Memory experiments

  • Memory experiments encode a known logical qubit, perform multiple error-correction cycles, and declare success when the decoded final outcome matches the initial logical state.Source: §A.1.1 · p. 28
  • The memory framework studies noise ranging from noiseless and code-capacity cases to phenomenological noise affecting physical qubits and readouts.Source: §A.1.1 · p. 28

A.1.2. The rotated surface code

  • The rotated surface code, introduced in as a surface-code variant of Kitaev’s toric code, places stabilizers in a checkerboard pattern on a two-dimensional data-qubit grid.Source: §A.1.2 · p. 28
  • Stabilizer ancilla qubits perform readouts through circuits that depend on the code variant, including separate X and Z readouts for the rotated surface code.Source: §A.1.2 · p. 28, Figure S1 · p. 29

A.1.3. Sycamore memory experiment dataset … A.1.7. Soft measurement inputs vs. soft event inputs

This section describes the Sycamore dataset and detector-error-model simulations, including SD6 and SI1000 noise assumptions and measurement-noise modeling. It also examines soft stabilizer-measurement and soft event inputs, their probabilistic relationship, and the effect of uncertainty on the corresponding XOR transformation.

A.1.3. Sycamore memory experiment dataset

  • The publicly released Sycamore surface-code dataset comprises four code-size 3 × 3 areas—north, south, east, and west—and one code-size 5 × 5 area.Each dataset was split into even and odd subsets for two-fold cross-validation, with a detector error model fitted to each subset for decoding the other.Source: §A.1.3 · p. 29

A.1.4. Detector error model (DEM)

  • Detector error models represent independent stochastic error mechanisms as hyperedges connecting the detector clusters they trigger, with associated error probabilities.The models were previously fitted to each experimental set using a generalisation of the p_ij method, and Stim generates samples for decoder pre-training.Source: §A.1.4 · p. 29

A.1.5. Circuit depolarizing noise: SD6 and SI1000

  • SI1000 models superconducting-circuit noise with non-uniform Pauli-error strengths, assigning measurements relative weight 5p and single-qubit gates and idling p/10.SD6 is defined as a standard depolarizing six-step circuit cycle, while SI1000 uses a 1000 ns superconducting-inspired cycle [34].Source: §A.1.5 · p. 30

A.1.6. Measurement noise

  • Measurement noise arises from environmental interaction, qubit reuse, state transitions, dephasing, and noisy analog readout, including leakage and decay between states.The simulations can use discrete assignment errors or one-dimensional analog signals parameterized by SNR and normalized duration t = t_meas/T_1; the network receives post1 and post2 probabilities, preserving thresholded outcomes and leakage information.Source: §A.1.6 · pp. 33–34

A.1.7. Soft measurement inputs vs. soft event inputs

  • Directly providing stabilizer measurements rather than stabilizer detection events benefits the model (Fig. S15).Traditional detection events are binary changes between consecutive stabilizer measurements, computed as d_i,n := s_i,n ⊕ s_i,n−1.Source: §A.1.7 · p. 36
  • Measurement and event inputs contain the same amount of information for the decoder and can therefore be compared through a bijection.Source: §A.1.7 · p. 36
  • Soft XOR transforms pairs of posterior measurement probabilities into detection-event probabilities through a linear recurrence that can be integrated in reverse.Source: §A.1.7 · p. 37
  • Complete uncertainty, p_i,n = 1/2, makes the transformation non-invertible because all information is lost.Source: §A.1.7 · p. 37
  • For measurement probabilities not equal to 1/2, thresholding each measurement and XOR’ing the resulting Boolean values is equivalent to iterative soft XOR followed by thresholding.Source: §A.1.7 · p. 37

A.1.8. Pitfalls for training on soft information … A.2.2. Experiment at a fixed number of cycles

The supplementary methods prevent label leakage from soft readouts, define realistic leakage-aware simulations, and specify logical-error metrics for both variable- and fixed-cycle experiments. The metric framework treats logical error per round as the parameter governing exponential decay and derives it consistently across experimental settings.

A.1.8. Pitfalls for training on soft information

  • Thresholding final-round data-qubit and leakage measurements prevents soft-readout inputs from revealing the logical measurement label.Recomputed soft stabilizers map d^2 floating-point values to d^2 − 1 values, enabling label inference; thresholding restores the information content of standard non-soft readouts.Source: §A.1.8 · p. 38

A.1.9. Pauli+ model for simulations with leakage

  • The Pauli+ simulator models realistic device noise with leakage states, correlated crosstalk, depolarizing channels, and leakage removal, targeting approximately Λ = 4 performance.Noise strengths are informed by device characterization and projected future-device improvements; leakage is introduced through CZ dephasing, heating, and crosstalk mechanisms.Source: §A.1.9 · pp. 38–39
  • The Pauli+ model tracks leaked qubit states alongside Pauli operators and converts general noise channels into Pauli-twirled or stochastic transitions.Two-qubit gates acting on leaked inputs are replaced by noise channels on the non-leaked qubit, while leakage levels are explicitly represented.Source: §A.1.9 · p. 39

A.2. Metrics

  • The metrics section connects decoder error rates across cycle counts to a common logical-error-per-round quantity for comparing experiments.The supplied metric definitions distinguish cycle-dependent error rates from the derived per-round parameter.Source: §A.2.1 · p. 40, §A.2.2 · p. 40

A.2.1. Logical error per round

  • The decoder error rate E(n) is modeled from zero at cycle 0 toward 1/2 at large n, with ε defining the logical error per round and exponential decay.The formulation gives E(1) = ε and identifies ε as the quantity governing the decay of logical-state fidelity.Source: §A.2.1 · p. 40
  • Estimating ε from experiments with different cycle counts uses compatible procedures, including a single-round fit constrained by F(0) = 1.The single-round procedure yields exactly the same logical error per round as the alternative derivation.Source: §A.2.1 · p. 40

A.2.2. Experiment at a fixed number of cycles

  • For a fixed-cycle experiment, such as n = 25, the logical error per round is obtained by inverting the logical-error model.This provides the fixed-cycle conversion from the observed error rate to ε.Source: §A.2.2 · p. 40

A.2.3. Experiment across multiple rounds … A.4.6. Efficient training for variable-duration experiments

This section evaluates fit quality and statistical aggregation and describes a recurrent decoder architecture tailored to surface-code decoding. It also examines auxiliary stabilizer-prediction tasks and efficient training for variable-duration experiments, including predictions at every cycle with fewer recurrent-network applications.

A.2.3. Experiment across multiple rounds

  • Fits for 3 × 3 and 5 × 5 memory experiments are assessed using R^2 and departures of log F_0 from 0, while excluding the first-cycle point because of a time-boundary effect.The excluded point is (1, E(1)), which otherwise produces much stronger apparent error suppression per round.Source: §A.2.3 · p. 40

A.3. Note on statistics

  • Aggregated uncertainty combines bootstrap estimates of individual fidelity points through Gaussian error propagation while excluding performance spread across datasets or, for multiple seeds, bootstrapped sample variance.Experimental aggregates contain 16 distinct datasets for distance 3 or 4 for distance 5; these vary by basis, subset, and device region.Source: §A.3 · p. 42

A.4. Model details

  • The recurrent architecture maintains a decoder state encoding prior stabilizers, allowing information storage over an arbitrary experiment duration and supporting multiple code distances.The architecture is designed around the time-invariant nature of decoding, with several features becoming more important at larger code distances.Source: §A.4 · p. 42

A.4.1. Input representation

  • The input representation supports binary events, measurements, probabilistic I/Q inputs, and leakage features, with measurement inputs performing better than event inputs in practice.Stabilizer-index embeddings, special final-round embeddings, residual processing, and 0.707 state scaling complete the representation pipeline.Source: §A.4.1 · pp. 42–43

A.4.2. Syndrome Transformer

  • The Syndrome Transformer combines self-attention, 2-dimensional convolutions, and dilated convolutions so local correlations and long-range stabilizer interactions can be modeled without rigid spatial invariance.Dense attention can reason over all stabilizer pairs, while convolutions capture local structure in the surface-code layout.Source: §A.4.2 · p. 43

A.4.3. Attention bias

  • The model learns spatial and temporal attention biases from stabilizer-pair layout and event-correlation features, and these biases can be precomputed after training.Visualized heads show distinct local, non-local, same-basis, and self-attention preferences, although interpretability is inconsistent across models.Source: §A.4.3 · pp. 43–44

A.4.4. Readout network

  • The readout scatters stabilizer states to data-qubit layouts, combines neighboring stabilizers with convolutions, pools perpendicular to logical observables, and predicts labels for equivalent logical-observable lines.Only the first line is used at inference time, although training can average losses across all available labels.Source: §A.4.4 · p. 45

A.4.5. Auxiliary tasks

  • The network is additionally trained to predict the next measurement as an auxiliary task, motivated by the possibility that auxiliary predictions improve training or main-task performance.The supplied passage specifies the auxiliary target but does not report its effect here.Source: §A.4.5 · pp. 46–47
  • The auxiliary stabilizer-prediction task slightly detracts from performance but leads to slightly faster training.The task uses a linear projection and logistic output unit from each stabilizer’s representation.Source: §A.4.5 · p. 47

A.4.6. Efficient training for variable-duration experiments

  • Variable-duration training shares simulator states and computation across experiments, while accounting for distinct final stabilizers at each duration.Repeated simulations or access to internal simulator states provide a shared bulk and a label for each round, with final stabilizers differing from bulk stabilizers.Source: §A.4.6 · p. 47
  • For N cycles, training N labels requires 2N embedding and RNN-core applications plus N Readout computations, versus N(N + 1)/2 recurrent applications for separate durations.The shared approach is illustrated in Fig. S7 and uses distinct bulk and final-stabilizer processing for each duration.Source: §A.4.6 · p. 47

A.5. Implementation details · A.6. Training details

The decoder is implemented and trained with JAX, Haiku, and JAXline, with an architecture that predicts labels at every round using stabilizer representations and experiment-basis information from final data-qubit measurements.

A.5. Implementation details

  • The ML decoder is implemented and trained using the JAX, Haiku, and JAXline machine-learning frameworks.Source: §A.5 · p. 47
  • The network architecture predicts labels at every round using stabilizer representations S_n.Source: Figure S7 · p. 48
  • The primed embeddings use different embedding parameters and only stabilizers in the experiment basis computed from final data-qubit measurements.These measurements are described as part of Appendix A.4.1.Source: §A.5 · p. 48

A.6.1. Sycamore data … A.7. Ensembling

The decoder is trained and evaluated through dataset-specific validation, pre-training, fine-tuning, auxiliary objectives, and development-set model selection. Training also uses distance-specific hyperparameters and parameters, while ensembling combines independently trained models for Sycamore and scaling experiments.

A.6.1. Sycamore data

  • The Sycamore pipeline performs 2-fold cross-validation, pre-trains on samples from a DEM fitted on half the data, and fine-tunes using the first 19 880 experimental samples.Source: §A.6.1 · pp. 48–49

A.6.2. SI1000 and I/Q noise

  • SI1000 experiments use Stim-generated data with I/Q noise, 25 error-detection cycles, five seeds, and models trained for 2 billion samples; I/Q models vary hard or soft inputs, SNR ∈{5, 10, 20}, and t∈{0, 0.003, 0.01, 0.03}.The SI1000 setting uses p=0.2%, and parameters are selected by the best fitted LER on the development set.Source: §A.6.2 · p. 49

A.6.3. Pauli+

  • Scaling experiments use Pauli+ data across code distances 3, 5, 7, 9, and 11, four leakage levels, hard or soft inputs, auxiliary prediction tasks, and post-processed I/Q noise.Each distance uses 990 million Pauli+ training samples and 2 billion total training samples, with the highest-development-set-LER parameters retained.Source: §A.6.3 · p. 50

A.6.4. Loss

  • The training objective averages binary-target cross-entropy losses across logical observables and down-weights auxiliary next-stabilizer prediction loss across cycles and stabilizers.Loss is minimized with stochastic gradient descent using Lamb for experimental data and Lion for scaling data, with weight decay, warm-up, scheduled learning-rate reductions, and batch size 256.Source: §A.6.4 · p. 51, §A.6.5 · p. 52

A.6.6. Termination

  • Training terminates after 2 billion examples; exponentially averaged parameters are regularly evaluated on development data for LER, with the lowest-development-set-LER parameters retained and evaluation specified across cycles.Training is not fully converged by 2 billion examples for larger code distances, producing higher validation-set LER than expected.Source: §A.6.6 · p. 52

A.6.7. Hyperparameters

  • Hyperparameters were tuned using 5 × 5 DEM training and validation data for Sycamore, further tuned on 11 × 11 validation LER for scaling, then reused across distances with learning-rate adjustments.The same architecture was used for all code distances.Source: §A.6.7 · pp. 52–53, §A.6.8 · p. 55

A.6.8. Parameters

  • All convolutions are 3×3, while dilation varies with code distance and larger distances add stabilizer-index embedding parameters.Table S5 reports the resulting parameter counts for each model.Source: §A.6.8 · p. 55

A.7. Ensembling

  • Ensembles combine independently trained models without changing computation speed because models run in parallel, though they require more computational resources.The authors use 20 seeds for Sycamore and 5 for scaling, average logits equivalently to taking a geometric mean of predicted error probabilities, and ensemble all trained models.Source: §A.7 · p. 55

A.8. Decoding speed … B. Supplementary text

The decoder is currently slower than PyMatching on available hardware, but its runtime has favorable noise and code-distance scaling, with parallelization offering further speed potential. Supplementary analyses also calibrate output confidence and define error-suppression factors for Pauli+ experiments.

A.8. Decoding speed

  • Inference speed could improve through distance-specific simplification, distillation, lower-precision inference, weight pruning, and specialized ASIC or FPGA implementations.Parallel operations and pipelining can increase throughput, potentially by trading higher latency for greater throughput.Source: §A.8 · pp. 55–57
  • At moderate noise levels, our decoder is slower on current hardware than needed for a practical quantum computer, but its code-distance scaling is better than PyMatching.Figure S9 shows decoding time per QEC cycle versus code distance.Source: §A.8 · p. 55, Figure S9 · p. 56

A.8.1. Decoding speed scaling considerations

  • The model runtime is independent of physical noise level, whereas matching slows as noise increases.Fixed runtime is identified as a practical advantage of neural-network decoders [95].Source: §A.8.1 · p. 57
  • The attention blocks have per-cycle computational cost scaling as approximately d^4, although the model components are intrinsically parallelizable.The realized scaling depends on execution hardware; sparse attention could reduce overall parallel compute requirements to approximately d^2.Source: §A.8.1 · p. 57
  • On generally available GPUs and TPUs through code distance d=25, the ML decoder achieves sub-quadratic scaling, suggesting a crossover with single-threaded MWPM at larger distances.This comparison remains challenging because matching algorithms have undergone extensive optimization, including approaches that avoid directly executing Blossom [44].Source: §A.8.1 · p. 57

A.9. Postselection and calibration

  • For code distances 3–11, the supplementary calibration analysis evaluates predicted probabilities under Pauli+ data with SNR=10, t=0.01, and 0.1% stabilizer-qubit leakage.Calibration histograms separate correct and incorrect predictions; incorrect predictions have lower certainty, while correct predictions cluster near probabilities 0 and 1.Source: Figure S10 · p. 58, Figure S11 · p. 58
  • The Pauli+ experiment’s average error-suppression factor is computed from Fig. 3B as Λ3/11=(ε3/ε11)1/4, using logical error per round at distances 3 and 11.For the 10B run, ε3 comes from the 2B run while ε11 comes from the 10B run because the model is well-converged at distance 3.Source: Table S6 · p. 59, §A.9 · p. 59

B.1. Further details of scaling experiments · B.2. Time Scalability · B.3. Soft matching

The supplementary experiments show that the ML decoder scales favorably across code distances, leakage strengths, and very long decoding sequences. They also describe recurrent decoding’s constant per-round resource use and a soft-information extension for MWPM, while identifying remaining comparison gaps.

B.1. Further details of scaling experiments

  • LER reaches 5.37 ± 0.01 through distance 11, with the ML decoder achieving the strongest reported error suppression.Table S6 collects error suppression factors across distances 3–5 and 3–11 and multiple decoders and input modalities.Source: §B.1 · p. 59
  • The ML decoder’s performance degrades significantly less as leakage inputs increase than PyMatching and MWPM-Corr.The comparison is shown in Fig. 3c, with detailed leakage-strength results in 25-cycle simulations at SNR = 10 and t = 0.01 for distances 3–11.Source: §B.1 · p. 59, Figure S13 · p. 60, Figure S12 · p. 60

B.2. Time Scalability

  • Networks trained for up to 25 cycles maintain decoding performance for experiments lasting up to 100 000 cycles or until fidelity falls below 10%.The longer-duration results are presented in Fig. S14.Source: §B.2 · p. 61, Figure S14 · p. 61
  • The observed LER decrease for matching and ML decoders at smaller distances is hypothesized to arise from relatively high simulated measurement noise.Terminal-round data-qubit measurements can create damaging space-like separated detection-event pairs, worsened when I/Q information is withheld; training used 25 cycles to mirror.Source: §B.2 · p. 61
  • The recurrent architecture keeps each error-correction step’s time and memory constant, so round 60 661 costs the same as round 7.PyMatching would instead require the entire matching graph as input, creating significant memory overhead; streaming-decoder comparisons remain future work.Source: §B.2 · pp. 61–62

B.3. Soft matching

  • Soft MWPM replaces average measurement-error probabilities with instance-specific posterior probabilities inferred from each I/Q sample.Thresholded posteriors produce binary outcomes and detection events, while opposite-outcome probabilities reweight error edges and can propagate through correlation reweighting [28].Source: §B.3 · p. 62
  • Soft matching withholds data-qubit posterior probabilities for fair comparison with the ML decoder, leaving leakage-aware matching comparison to future work [87].The unresolved leakage-aware approach would reweight edges based on leakage detections.Source: §B.3 · p. 62

B.4. Ablations · B.4.1. Model Ablations · B.4.2. Input Ablations

The ablations test how architectural and input components affect decoder performance under fixed hyperparameters and multiple random seeds. Effects are generally small at 5 × 5 but more pronounced at 11 × 11, with results reported for both performance and data efficiency.

B.4. Ablations

  • The ablation study removes or simplifies one architectural component at a time while leaving other hyperparameters unchanged.Performance can potentially be recovered by compensating with other changes, so these variants isolate effects under a fixed setup rather than optimized alternatives.Source: §B.4 · p. 62
  • Ablation effects are generally small at 5 × 5 but more marked at 11 × 11 across the tested decoder variants.Each scenario uses five models with different random seeds, comparing mean test-set LER for 5 × 5 Sycamore DEM pre-training and 11 × 11 Pauli+ training.Source: §B.4 · p. 62

B.4.1. Model Ablations

  • Model ablations replace or simplify the recurrent core, convolutions, residual stacks, pooling, attention bias, prediction loss, dimensions, or Transformer depth.The LSTM variant uses six LSTMs with distance-dependent hidden width and removes scatter and pooling, while other variants remove convolutional elements or reduce layer counts.Source: §B.4.1 · pp. 62–63
  • Additional model variants remove two-dimensional stabilizer pooling, attention-bias features, or next-stabilizer prediction, and reduce stabilizer dimensions or Syndrome Transformer layers.PoolingStabs pools across all stabilizers instead of along data-qubit rows or columns; FewerLayers reduces the Syndrome Transformer from three layers to one.Source: §B.4.1 · p. 63

B.4.2. Input Ablations

  • Input ablations separately provide syndrome information as detection events only or as raw qubit measurements only.These variants remove measurements or events, respectively, to test the contribution of each input representation.Source: §B.4.2 · p. 63
  • Figure S15 compares ablated and non-ablated decoder performance, averaging 5 × 5 DEM results across bases and cross-validation folds and reporting 11 × 11 Pauli+ results across five runs.The figure orders variants by mean LER and includes uncertainty estimates; the 5 × 5 panel also marks the fine-tuned ensemble performance.Source: Figure S15 · p. 64, §B.4.2 · p. 64
  • Figure S16 evaluates performance and data efficiency for the best-performing ablation subsets in 5 × 5 DEM and 11 × 11 Pauli+ settings.Its colors indicate the number of training samples needed to reach performance parity with PyMatching in 5 × 5 DEM.Source: Figure S16 · p. 65, §B.4.2 · p. 65

B.5. Choice of data and noise model for pre-training

Matched device-experimental noise models substantially improve pre-training over simpler SD6 and SI1000 models, with benefits persisting after fine-tuning. Fine-tuning on real experimental data nevertheless keeps the neural decoder competitive without a carefully tuned noise model.

B.5. Choice of data and noise model for pre-training

  • Because fine-tuning used very little data, more experimental data or a better training noise model could further improve decoding performance.Obtaining a hardware-matched DEM requires specialist effort, while SI1000 and SD6 are simpler but less expressive alternatives.Source: §B.5 · p. 66
  • The study pre-trained distance-3 and distance-5 models with SI1000 and SD6, compared them with DEM-pretrained models, and fine-tuned each on limited experimental data.DEM models used 2 billion examples and 20 seeds, whereas SI1000 and SD6 training stopped after 500 million examples and used 5 seeds; matched event densities were p = 0.006 and p = 0.008.Source: §B.5 · p. 66
  • Fine-tuning models pretrained with coarser noise brings performance close to the tensor-network decoder, keeping the neural decoder competitive without a carefully tuned noise model.The results also underscore the importance of fine-tuning on real experimental data.Source: §B.5 · p. 66
  • Pre-training on matched DEM data outperforms SD6 and SI1000, and this advantage persists after fine-tuning on experimental data.Figure S17 compares pre-training with experimental-fitted DEM, SD6, and SI1000 noise, with optional fine-tuning on experimental data.Source: §B.5 · p. 66, Figure S17 · p. 67

B.6. Training Data Efficiency

The decoder’s required training-data scale remains manageable with increasing code distance, achieving parity with matching decoders using far fewer samples than prior work at distances 9 and above [95]. The authors also propose distributed decoding as a compatible strategy while retaining flexibility for nonlocal error mechanisms.

B.6. Training Data Efficiency

  • At code distances 9 and above, the architecture reaches parity with PyMatching and MWPM-Corr using several magnitudes fewer training samples than prior work [95].Figure S18 estimates the required sample size by the first crossover with these matching decoders.Source: §B.6 · p. 67, Figure S18 · p. 68
  • A distributed decoder could apply deep learning locally and combine it with a deterministic global decoder such as union find [64] [96].This parallelism resembles the CNN- and RNN-based parameter sharing used in the presented decoder.Source: §B.6 · p. 67
  • Unlike locality-assuming distributed schemes, the approach accommodates a more flexible range of correctable errors without assuming locality in the error mechanism.This flexibility is identified as an advantage of the architecture’s CNN and RNN parameter sharing.Source: §B.6 · p. 67
  • Figure S18 measures training efficiency using five distance-11 training curves and average first-crossover points across code distances.The curves use Pauli+ generated data with SNR = 10, 𝑡= 0.01, and 0.1% leakage, with crossover points marking parity against PyMatching or MPWM-Corr with soft inputs.Source: §B.6 · pp. 67–68, Figure S18 · p. 68

Related papers

Loading 2310.05900v1…