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arxiv: 2312.03508 · v2 · submitted 2023-12-06 · 🪐 quant-ph

Convolutional neural network based decoders for surface codes

Simone Bordoni , Stefano Giagu This is my paper

Pith reviewed 2026-05-24 05:09 UTC · model grok-4.3

classification 🪐 quant-ph
keywords surface codesquantum error correctionconvolutional neural networksdecodersnoise modelsexplainable machine learningsyndrome decoding
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The pith

Convolutional neural network decoders achieve good performance on surface codes and adapt to different noise models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests decoders built from convolutional neural networks on surface code error syndromes across multiple code distances and several noise models. It shows these networks maintain solid correction rates while shifting between noise conditions without major redesign. The work also applies explainable machine learning tools to inspect the network's internal decisions and error patterns, with the goal of refining the decoder for greater reliability. If the results hold, decoding could stop acting as a bottleneck that slows quantum operations when noise characteristics change during a run.

Core claim

The authors report that convolutional neural network decoders for surface codes deliver good performance on error syndrome data drawn from different code distances and from multiple noise models. They further show that the same network architecture can be retrained or adapted to handle shifts in the underlying noise statistics. Explainable machine learning analysis is used to map which input features drive the decoder's outputs and where mistakes arise, providing a route to iterative improvements in robustness.

What carries the argument

convolutional neural network trained to map surface-code syndrome patterns to error corrections, with the network weights adjusted to different noise statistics

If this is right

  • Decoding time no longer scales with classical algorithm complexity when code distance grows.
  • A single decoder architecture can be reused across runs that experience different noise environments.
  • Explainable analysis of network decisions can guide targeted retraining to reduce specific error classes.
  • Surface-code logical qubits can be operated at higher effective rates if syndrome processing keeps pace with hardware cycle times.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hardware implementations could embed the network weights directly in control electronics to remove data transfer latency between quantum device and classical processor.
  • Periodic retraining on fresh calibration data might allow the decoder to track slow drifts in qubit parameters without interrupting computation.
  • The same convolutional approach could be tested on other topological codes whose syndrome graphs share local structure with the surface code.

Load-bearing premise

The simulated noise models and training distributions match the actual error processes that occur on physical quantum hardware.

What would settle it

Running the trained CNN decoder on a real quantum device and measuring logical error rates that deviate sharply from the simulated predictions under the same nominal noise parameters.

Figures

Figures reproduced from arXiv: 2312.03508 by Simone Bordoni, Stefano Giagu.

Figure 1
Figure 1. Figure 1: Schematic representation of a surface code. Data qubits are represented as circles [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Examples of logical operators in a d = 5 surface code. Z and X logical opera￾tors are composed of chains of single qubit Z or X operators connecting the Z or X sides respectively. 2.1 Errors Unlike the classical bit, where only bit flip errors may appear, for a qubit there is a continuous set of possible errors. However, after a measurement cycle, the state of the qubits collapses into a discrete set of po… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between accuracy obtained with HLD based on FFNN and CNN and [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Performance of MWPM for the de￾polarising plus measurement errors model us￾ing a different number or imperfect measure￾ment cycles (3 and 9). The depolarising error probability (p) has been set equal to the mea￾surement error probability (q). For this noise model a different number of measurement cy￾cles does not modify significantly the decoding problem. cycle is performed. The measurement error probabili… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison between accuracy obtained with HLD based on CNN and the accuracy [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the accuracy, on dif￾ferent code distances (d), obtained with a CNN with no dilation factor (N) and the same model with a dilated convolution with a dilation rate equal to two on all the convolutional layers except the first one (D). For the depolarising error model (Top) the neural networks have been trained on a dataset with p = 0.1. While for the depolarising plus measurement errors model … view at source ↗
Figure 7
Figure 7. Figure 7: The results obtained in this study show that dilated convolution may be a good way to im￾prove the performance of the decoder for high distance codes. In fact, while for codes of distance 7 and 9 the use of dilated convolu￾tion doesn’t alter the performance sensibly, for codes of distance 11, where there are more con￾volutional layers, it is possible to obtain a per￾formance improvement. The performance in… view at source ↗
Figure 8
Figure 8. Figure 8: Accuracy of the HLD trained and tested on different single qubit error probabilities [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Accuracy of the HLD trained and tested on different depolarising and measurement [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Saliency map of a manually gen￾erated error syndrome analysed with a neural network decoder, the dark red regions are the ones that contribute more to the output. The error syndrome has been overlaid to the map, data qubits with X errors are highlighted in green, while the measurement qubits that re￾port the error signal are highlighted in blue. Two single X errors, E1 and E2, and two er￾rors chains compo… view at source ↗
Figure 11
Figure 11. Figure 11: Saliency map of a manually gen￾erated error syndrome analysed with a neural network decoder, the dark red regions are the ones that contribute more to the output. The error syndrome has been overlaid to the map, data qubit with depolarising errors are high￾lighted in green, while the measurement qubit that report the error signal are highlighted in blue. Two errors chains composed respectively of five X (… view at source ↗
Figure 12
Figure 12. Figure 12: Saliency map of an error syndrome analysed with a neural network decoder, the dark red regions are the ones that contribute more to the output. The error syndrome has been overlaid to the map, data qubit with X errors are highlighted in green, while the mea￾surement qubit that report the error signal are highlighted in blue. In this case the HLD is able to correct the error syndrome while the MWPM fails. … view at source ↗
Figure 13
Figure 13. Figure 13: Performance comparison between HLD trained on a standard train set with single [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: A comparison with Fig. 11 shows how [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 14
Figure 14. Figure 14: Saliency map of the same error syn￾drome reported in [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
read the original abstract

The decoding of error syndromes of surface codes with classical algorithms may slow down quantum computation. To overcome this problem it is possible to implement decoding algorithms based on artificial neural networks. This work reports a study of decoders based on convolutional neural networks, tested on different code distances and noise models. The results show that decoders based on convolutional neural networks have good performance and can adapt to different noise models. Moreover, explainable machine learning techniques have been applied to the neural network of the decoder to better understand the behaviour and errors of the algorithm, in order to produce a more robust and performing algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates convolutional neural network (CNN) based decoders for surface code error syndromes. It evaluates performance on varying code distances and noise models (including depolarizing and bit-flip), asserts that the decoders achieve good performance and adapt to different noise models, and applies explainable machine learning techniques to analyze network behavior and errors for improved robustness.

Significance. If the performance and adaptation results are substantiated by rigorous quantitative benchmarks against established decoders, this could support development of efficient, flexible decoders relevant to fault-tolerant quantum computation. The application of explainable ML techniques is a positive feature that may aid interpretability and refinement of ML-based quantum decoders.

major comments (2)
  1. [Abstract and results] Abstract and results sections: performance and adaptation claims are stated without reported numerical error rates, thresholds, baselines (e.g., MWPM or union-find decoder comparisons), error bars, or training/validation split details. This is load-bearing for the central claim that the CNN decoders 'have good performance and can adapt to different noise models'.
  2. [Results and discussion] Results and discussion on noise models: adaptability is demonstrated only within fixed phenomenological or circuit-level simulated models; no transfer metrics, cross-model generalization tests, or validation against hardware-calibrated or non-Markovian noise distributions are provided. This directly affects the hardware relevance of the adaptation result.
minor comments (2)
  1. [Methods] Provide explicit description of the CNN architecture, layer dimensions, activation functions, loss function, and optimizer in the methods section for reproducibility.
  2. [Figures] Ensure all figures include clear legends, axis labels, and captions that allow direct comparison of CNN performance to baselines across code distances.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable feedback. We address each major comment below and outline the revisions we will make to improve clarity and substantiation of the claims.

read point-by-point responses

  1. Referee: [Abstract and results] Abstract and results sections: performance and adaptation claims are stated without reported numerical error rates, thresholds, baselines (e.g., MWPM or union-find decoder comparisons), error bars, or training/validation split details. This is load-bearing for the central claim that the CNN decoders 'have good performance and can adapt to different noise models'.

    Authors: We agree that explicit numerical reporting strengthens the manuscript. The performance comparisons are visualized in the results figures (logical error rates vs. physical error rate for varying distances and noise models), but we will add a dedicated table in the revised results section listing specific error rates at key points, estimated thresholds where observable, direct numerical comparisons to MWPM and union-find baselines, standard error bars from repeated training runs, and the precise training/validation/test split ratios used. This addresses the load-bearing aspect of the central claim without altering the underlying data. revision: yes

  2. Referee: [Results and discussion] Results and discussion on noise models: adaptability is demonstrated only within fixed phenomenological or circuit-level simulated models; no transfer metrics, cross-model generalization tests, or validation against hardware-calibrated or non-Markovian noise distributions are provided. This directly affects the hardware relevance of the adaptation result.

    Authors: The manuscript demonstrates adaptation by retraining the same CNN architecture independently on each noise model (phenomenological depolarizing and bit-flip) and obtaining competitive performance in each case. We acknowledge that this does not include cross-model transfer metrics or tests on hardware-calibrated/non-Markovian noise. In revision we will expand the discussion section to explicitly state the scope (simulated Markovian models only), add a limitations paragraph on the absence of transfer learning experiments, and note that hardware validation remains future work. No new experiments will be added at this stage. revision: partial

Circularity Check

0 steps flagged

No circularity: standard empirical ML evaluation on simulated data

full rationale

The paper applies convolutional neural networks to surface-code decoding via supervised training on simulated syndromes. No derivation chain, equations, or first-principles claims exist that reduce to self-definition, fitted inputs renamed as predictions, or self-citation load-bearing steps. Performance and adaptability results are obtained by training on one noise model and evaluating on held-out data or different models, which is externally falsifiable against the simulations and does not collapse by construction. The central claims rest on empirical benchmarks rather than any internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions of quantum error correction and supervised machine learning; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Surface-code syndromes contain sufficient information to identify correctable errors under the assumed noise models
    Standard premise of quantum error correction decoding.
  • domain assumption Supervised training on simulated data produces a decoder that generalizes to the target noise distribution
    Core assumption of any machine-learning decoder trained on synthetic data.

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discussion (0)

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