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Scalable Neural Decoder for Topological Surface Codes

Kai Meinerz, Chae-Yeun Park, Simon Trebst

arXiv:2101.07285v2quant-phcond-mat.stat-mech

TL;DR

Quantum error correction needs decoders that can scale while decoding quickly, but widely used approaches face speed or general-noise threshold limitations. This paper combines local neural-network preprocessing with union-find decoding, reporting higher thresholds and faster decoding across a range of error rates. Its measured scaling reaches about 130,000 qubits, though GPU initialization and data-loading overhead affect smaller systems.

  • Problem

    Quantum error correction needs efficient syndrome decoders, while MWPM implementations can be too slow and its near-optimal threshold property does not extend to more general noise models.

    Source: p. 1

  • Method

    A hierarchical decoder uses a sliding local neural-network preprocessing window to correct locally inferable errors before union-find handles remaining longer-range errors.

    Source: pp. 1–2

  • Results

    The ML-assisted decoder reaches a depolarizing-noise threshold of 0.162(5), 10% above the union-find threshold of 0.146(1).

    Source: p. 3

  • Takeaways & Limitations

    The approach combines increased logical accuracy and thresholds with reduced decoding times, complementing lazy union-find, which excels at ultralow error rates.

    Source: p. 5

    GPU initialization and syndrome transfer impose near-constant overhead that produces a performance plateau at small system sizes.

    Source: p. 4

Abstract

from arXiv · show

With the advent of noisy intermediate-scale quantum (NISQ) devices, practical quantum computing has seemingly come into reach. However, to go beyond proof-of-principle calculations, the current processing architectures will need to scale up to larger quantum circuits which in turn will require fast and scalable algorithms for quantum error correction. Here we present a neural network based decoder that, for a family of stabilizer codes subject to depolarizing noise and syndrome measurement errors, is scalable to tens of thousands of qubits (in contrast to other recent machine learning inspired decoders) and exhibits faster decoding times than the state-of-the-art union find decoder for a wide range of error rates (down to 1%). The key innovation is to autodecode error syndromes on small scales by shifting a preprocessing window over the underlying code, akin to a convolutional neural network in pattern recognition approaches. We show that such a preprocessing step allows to effectively reduce the error rate by up to two orders of magnitude in practical applications and, by detecting correlation effects, shifts the actual error threshold, up to fifteen percent higher than the threshold of conventional error correction algorithms such as union find or minimum weight perfect matching, even in the presence of measurement errors. An in-situ implementation of such machine learning-assisted quantum error correction will be a decisive step to push the entanglement frontier beyond the NISQ horizon.

Appendix A: Neural Networks

The appendix overviews the tested network architectures and training hyperparameters.

  • The appendix details tested network architectures and neural-network training hyperparameters.Source: Appendix A · p. 6

Network Architecture

The authors explore network architectures to balance decoding speed, error-threshold performance, and scalability. A 5×5 network is favored for speed, while a 7×7 network yields the highest threshold; the speed-optimized choice retains a strong threshold.

Network Architecture

  • The tested architectures vary subsystem size, hidden-layer count, and nodes per layer to optimize wall-clock decoding time.Source: Network Architecture · p. 6, Table II · p. 6
  • With syndrome measurement errors, the decoder uses a three-dimensional adaptation with a 3×3×3 octahedral subsystem.The appendix caption identifies this as the effectively three-dimensional setting.Source: Table III · p. 7, Figure 6 · p. 7
  • For speed, the best configuration uses a 5×5 subsystem, 3 hidden layers, and 128 nodes per layer, with 40,068 adjustable parameters.Source: Network Architecture · p. 7
  • For decoding efficiency, the most favorable configuration uses a 7×7 subsystem, 5 hidden layers, and 512 nodes per layer, with more than 10^6 adjustable parameters.Source: Network Architecture · p. 7
  • The speed-optimized 5×5 network has threshold p_th = 0.162(5), close to the efficiency-optimized 7×7 network’s maximum p_th = 0.167(0).The authors characterize the 5×5 configuration as a compromise between fast decoding, high thresholds, and scalability.Source: Network Architecture · p. 7

Training

The neural networks are trained with supervised syndrome–error examples generated separately for each error rate, with fresh examples produced each epoch.

Training

  • Each network uses supervised training on labeled syndrome–error pairs, with a separate network trained for each error rate.Source: Training · p. 7
  • Each epoch generates 512 new training pairs on the fly, making the effective dataset size roughly batch size × training epochs and reducing overfitting risk.Source: Training · p. 7
  • Training hyperparameters are grid-searched for the decoder’s error threshold and then reused across error probabilities.Source: Training · p. 7

Appendix B: Benchmarking

The benchmarking appendix specifies the CPU/GPU hardware used for timing measurements and explains how GPU kernel-launch time is included.

Appendix B: Benchmarking

  • Depolarizing-noise benchmarks use an Intel Xeon E5-2699A v4 CPU and Nvidia Tesla V100 SXM2 GPU.Source: Appendix B · p. 7
  • Benchmarks with depolarizing noise plus syndrome measurement errors use an AMD EPYC Rome 7402 CPU and Nvidia A100 Tensor Core GPU.Source: Appendix B · p. 8
  • Reported times cover decoding a syndrome and applying its correction, with GPU kernel-launch time included for machine-learning steps.Kernel time is estimated by running a neural network with subsystem ℓ=7 at perr=0.01, where launch time dominates data transfer and computation.Source: Appendix B · p. 8

Appendix C: Minimum Weight Perfect Matching (MWPM)

The appendix reviews MWPM complexity and locality-based implementations, then reports thresholds for PyMatching and its use within the hierarchical decoder. The hierarchical approach raises the reported threshold, while the authors retain UF for its scalability.

Appendix C: Minimum Weight Perfect Matching (MWPM)

  • The original Blossom algorithm is O(|E||V|2), or O(n4) in the toric-code setup; improved algorithms give O(n3) as the considered worst-case complexity.PyMatching's efficiency also uses locality, connecting only nearby defects; its benchmarks can achieve O(L2.11).Source: Appendix C · p. 8
  • The hierarchical ML-assisted MWPM threshold is perr = 0.1671 ± 0.0005, about 10% above bare MWPM; the authors nevertheless choose UF for scalability.Figure 7 compares conventional and ML-assisted MWPM across depolarized noise and noise with syndrome measurement errors.Source: Figure 7 · p. 8, Appendix C · p. 8
  • Figure 7 shows a roughly 10% threshold shift from perr = 0.1542 for bare MWPM to perr = 0.1671 for ML-assisted MWPM.The figure covers depolarized noise and an augmented model with syndrome measurement errors.Source: Figure 7 · p. 8
  • PyMatching's threshold is perr = 0.1542±0.0006, and nearby-edge construction barely shifts the threshold.The comparison is with the independent-noise estimate 3/2 · pindep, where pindep = 0.1031 ± 0.0001.Source: Appendix C · p. 8

Appendix D: Union-find (UF) decoder

The appendix describes the union-find decoder's cluster-growth and peeling procedure and reports its scaling and threshold behavior. It also notes that residual errors after ML preprocessing make the UF stage's runtime differ from bare UF at the corresponding effective error rate.

Appendix D: Union-find (UF) decoder

  • For 7 × 7 subsystems, the ML-assisted UF threshold is perr = 0.167(0), with critical exponent ν = 1.5(2).The larger subsystem slightly shifts the threshold relative to the more time-efficient 5 × 5 subsystem.Source: Figure 8 · p. 9
  • The UF decoder shows almost perfect linear scaling with lattice size over a wide range of error rates, with deviations near threshold.The authors attribute the deviations they observe near threshold to rehashing operations in their hash-set implementation.Source: Figure 9 · p. 9, Appendix D · p. 9
  • UF iteratively grows and merges clusters until each has an even number of defects, then uses peeling to infer corrections.The implementation uses a disjoint-set structure and boundary-vertex updating scheme to achieve O(nα(n)) scaling.Source: Appendix D · p. 9
  • With ML preprocessing, residual errors contain substantial long-range corrections, requiring more Grow and Merge operations; for larger systems the UF stage dominates overall decoding time.The authors do not expect its runtime to match bare UF at the corresponding effective error rate.Source: Appendix D · p. 9
  • The appendix presents lazy-UF threshold and scaling results for depolarizing noise and for noise augmented with syndrome measurement errors.The supplied text identifies the figure's scope but gives no numerical outcome.Source: Appendix D · p. 9

Appendix E: Literature overview

The appendix provides a literature overview to compare conventional and ML-assisted decoders by threshold and scaling. It also points to lazy-UF threshold and scaling results for two noise settings.

Appendix E: Literature overview

  • The overview is intended to support comparison of achievable thresholds and algorithmic scaling across decoder literature.Source: Appendix E · p. 9
  • Figure 10 presents lazy-UF threshold and scaling behavior for depolarizing noise and for an augmented model with syndrome measurement errors.The supplied caption does not report numerical results.Source: Figure 10 · p. 9
  • Table V compares decoder characteristics across toric, surface, semion, and color codes, listing noise model, threshold, largest tested code distance, and scaling in total qubits.Asterisks mark MWPM and UF threshold values determined in this work.Source: Table V · p. 10
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