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arxiv: 2506.16113 · v1 · submitted 2025-06-19 · 🪐 quant-ph

Fully convolutional 3D neural network decoders for surface codes with syndrome circuit noise

Spiro Gicev , Lloyd C. L. Hollenberg , Muhammad Usman This is my paper

Pith reviewed 2026-05-19 08:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords surface code decodingneural network decoderquantum error correctioncircuit-level noisefully convolutional networkrotated surface codeminimum weight perfect matching
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The pith

A fully convolutional 3D neural network decoder generalizes to rotated surface codes up to distance 97 with thresholds competitive to minimum-weight perfect matching.

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

The paper develops a vectorised simulation method and trains fully convolutional 3D networks on syndrome data from surface codes that include noise after every gate and idle step. These networks treat decoding as a multi-label classification task and are shown to scale from small training instances to rotated codes as large as distance 97. Thresholds reach 0.7 percent depolarising error, matching the performance of minimum-weight perfect matching while delivering lower latency once code distance exceeds 33 above threshold or 89 below threshold. The results indicate that such networks can meet the decoding speed and accuracy demands projected for fault-tolerant quantum computation.

Core claim

The authors show that a fully convolutional 3D neural network, trained on spatiotemporal syndrome tensors generated from rotated surface codes with circuit-level depolarising noise, generalises without retraining to distances up to d=97, attains depolarisation thresholds of 0.7 percent, and matches or exceeds the logical error rate of minimum-weight perfect matching while reducing wall-clock decoding time for distances d=33 and larger.

What carries the argument

The fully convolutional 3D neural network that ingests the full space-time syndrome volume as input and outputs correction decisions directly.

If this is right

  • Decoding latency drops below that of standalone minimum-weight perfect matching once distance reaches 33 under above-threshold noise.
  • Latency gains extend to distances of 89 and above under below-threshold noise.
  • The same trained network maintains competitive thresholds across the full range of circuit noise strengths examined.
  • Vectorised syndrome generation enables rapid creation of training data at the scale needed for large codes.
  • The approach meets the speed and accuracy targets required by current fault-tolerant resource estimates.

Where Pith is reading between the lines

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

  • Combining the neural network front-end with a lightweight residual classical decoder could push usable distances even higher.
  • The vectorised simulation technique may transfer directly to other stabilizer codes that possess regular lattice structure.
  • Hardware implementations could exploit the fixed-weight convolutional layers for low-latency inference on FPGA or ASIC platforms.

Load-bearing premise

Training the network only on smaller codes and limited noise samples produces reliable performance on much larger distances and complete circuit noise without overfitting or threshold collapse.

What would settle it

A measured threshold below 0.5 percent or a clear rise in logical error rate when the same network weights are applied to a distance-97 rotated surface code under full circuit noise.

Figures

Figures reproduced from arXiv: 2506.16113 by Lloyd C. L. Hollenberg, Muhammad Usman, Spiro Gicev.

Figure 1
Figure 1. Figure 1: A distance 5 rotated surface code logical qubit with examples of an X and Z logical operator shown with chains of blue and red labels, respectively. Measurement circuits of X and Z surface code stabiliser operators are shown to the right. Data qubits are shown in grey. X and Z stabiliser operators are shown with yellow and green tiles, respectively. where Vi and Pi are vertex and plaquette operators associ… view at source ↗
Figure 2
Figure 2. Figure 2: Visualising the decoding demands of surface codes. (a) shows a flowchart describing the classical processing demands of rotated surface codes during a quantum memory benchmark. (b) shows a space-time volume showing errors and syndrome changes associated with one quantum memory benchmark instance (time moves vertically upward). Blue and red lines are usually associated usually with X and Z data qubit errors… view at source ↗
Figure 3
Figure 3. Figure 3: A distance-5 rotated surface code represented using an array of unit cells (offset for clarity). Each unit cell may contain up to four data qubits (grey), two X stabiliser generators (yellow), and two Z stabiliser generators (green). Code boundaries are well defined by specifying the presence of qubits by eight bits per unit cell. Unused data qubits are shown in light grey. through CNOT gates using well kn… view at source ↗
Figure 4
Figure 4. Figure 4: Surface code unit cell syndrome measurement with highlighted error detecting regions and reference time steps as black bars. Gates act between qubits and their nearest neighbours, which sometimes are in adjacent cells. At boundaries, gates scheduled on absent qubits are not applied. The red region highlights a data qubit initialisation round. The yellow region highlights a data qubit idling round. The gree… view at source ↗
Figure 5
Figure 5. Figure 5: Surface code error bits propagating backwards from before the second CNOT round to before the first CNOT round. Error bits are written in row vectors of form [EX, EZ], where EX and EZ are X and Z error bits, respectively. space-time error bits associated with the respective qubits. Next, four random floats are sampled per unit cell to find the contribution from ancilla qubit preparation and measurement err… view at source ↗
Figure 6
Figure 6. Figure 6: The application of simplifier operators on rotated surface code data. The subfigure titled “Original Data” shows errors and syndrome bit changes of a distance 15 code with p ′ = 0.005 noise. The subfigure titled “Simplifiers” shows error chain loops identified to reduce total error count or break symmetry. The subfigure titled “Modified Data” shows the resultant set of edges after applying simplifiers [PI… view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Performance of DEM PyMatching for rotated surface codes suffering uniform depolarisation parameter, p ′ , noise. be justified on a real quantum device, although ANN-based training can also suffer the same pitfall. Additionally, the performance of PCM PyMatching is still of relevance as a point of comparison when it is used as a global decoder for ANN-based decoding. An interesting property of circuit noise… view at source ↗
Figure 9
Figure 9. Figure 9: Matrices showing intersections between LER curves for rotated surface codes of distances between 5 and 33 when decoded with PCM PyMatching. Text in each cell shows the depolarisation parameter value rounded to three decimal places based on extrapolation of local error rates calculated at depolarisation parameters at the nearest two tenth of a percent. 5 9 13 17 21 25 29 33 Distance 5 9 13 17 21 25 29 33 Di… view at source ↗
Figure 10
Figure 10. Figure 10: Matrices showing intersections between LER curves for rotated surface codes of distances between 5 and 33 when decoded with DEM PyMatching. Text in each cell shows the depolarisation parameter value rounded to three decimal places based on extrapolation of local error rates calculated at depolarisation parameters at the nearest two tenth of a percent [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Performance multi-label classifier ANN decoding, followed by PCM PyMatching for rotated surface codes suffering uniform depolarisation parameter, p ′ , noise [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Performance multi-label classifier ANN decoding, followed by DEM PyMatching for rotated surface codes suffering uniform depolarisation parameter, p ′ , noise. 2.3.3. Simplified Multi-label Classifier Decoder An identical neural network structure was trained on data having gone through the error simplification process described in Section 2.2.3. The use of error simplification reduces the negative impacts … view at source ↗
Figure 13
Figure 13. Figure 13: Output predictions of ANNs when trained on unsimplified data compared to simplified data. The original data corrections to circuit noise with depolarisation parameter p = 0.005 [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Performance multi-label classifier ANN (simplified data during training) decoding, followed by PCM PyMatching for rotated surface codes suffering uniform depolarisation parameter, p ′ , noise. 3 × 10 3 4 × 10 3 6 × 10 3 Depolarisation parameter, p 0 10 3 10 2 10 1 Logical X error rate y=x d5 d9 d13 d17 d21 d25 d29 d33 3 × 10 3 4 × 10 3 6 × 10 3 Depolarisation parameter, p 0 10 3 10 2 10 1 Logical Z error … view at source ↗
Figure 15
Figure 15. Figure 15: Performance multi-label classifier ANN (simplified data during training) decoding, followed by DEM PyMatching for rotated surface codes suffering uniform depolarisation parameter, p ′ , noise [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Output predictions a diffusion ANN after 1, 11 and 21 passes through the model. Two shots are shown, corresponding to different random noise given as input. The original data corrections to circuit noise with depolarisation parameter p ′ = 0.005 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Performance 11-pass diffusion ANN decoding, followed by PCM PyMatching for rotated surface codes suffering uniform depolarisation parameter, p ′ , noise. 3 × 10 3 4 × 10 3 6 × 10 3 Depolarisation parameter, p 0 10 3 10 2 10 1 Logical X error rate y=x d5 d9 d13 d17 d21 d25 d29 d33 3 × 10 3 4 × 10 3 6 × 10 3 Depolarisation parameter, p 0 10 3 10 2 10 1 Logical Z error rate y=x d5 d9 d13 d17 d21 d25 d29 d33 … view at source ↗
Figure 18
Figure 18. Figure 18: Performance 11-pass diffusion ANN decoding, followed by DEM PyMatching for rotated surface codes suffering uniform depolarisation parameter, p ′ , noise. respectively. We find that comparable performance is achieved compared to the models based on multi-label classification. As a final comparison between all three ANN decoders presented here, the intersection depolarisation parameters between difference d… view at source ↗
Figure 19
Figure 19. Figure 19: Matrices showing intersections between LER curves for rotated surface codes of distances between 5 and 33 when decoded with PCM or DEM PyMatching (MWPM) alone and the three ANN decoders presented in this work. Text in each cell shows the depolarisation parameter value rounded to three decimal places based on extrapolation of local error rates calculated at depolarisation parameters at the nearest two tent… view at source ↗
Figure 20
Figure 20. Figure 20: Decoding times for PCM PyMatching decoding compared with multi-label ANN decoding with PCM PyMatching for X and Z basis memory experiments with circuit noise depolarisation parameters near threshold, p ′ th ≈ 0.006. 0.00 0.05 0.10 0.15 0.20 0.25 Time/Shot (s) ANN+PCM Mop-up p 0 = 0.004 PCM PyMatching p 0 = 0.004 ANN+DEM Mop-up p 0 = 0.004 DEM PyMatching p 0 = 0.004 ANN Part (PCM) p 0 = 0.004 ANN Part (DEM… view at source ↗
Figure 21
Figure 21. Figure 21: Decoding times for PCM PyMatching and DEM PyMatching decoding compared with multi-label ANN decoding with the same decoders used for mop-up for Z basis memory experiments with uniform depolarisation parameter circuit noise. Results are shown for depolarisation parameters p ′ = 0.004 and p ′ = 0.005, corresponding to below-threshold performance [PITH_FULL_IMAGE:figures/full_fig_p026_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Relative matching decoding times for PCM PyMatching decoding compared with multi-label ANN decoding with PCM PyMatching for X and Z basis memory experiments with depolarisation parameter, p ′ , circuit noise. 2.5. Comparison With Previous Works The most direct comparison of our work to previous research on ANN decoding at large distances is with Meinerz et al. [58] and Chamberland et al. [45]. A summary i… view at source ↗
read the original abstract

Artificial Neural Networks (ANNs) are a promising approach to the decoding problem of Quantum Error Correction (QEC), but have observed consistent difficulty when generalising performance to larger QEC codes. Recent scalability-focused approaches have split the decoding workload by using local ANNs to perform initial syndrome processing and leaving final processing to a global residual decoder. We investigated ANN surface code decoding under a scheme exploiting the spatiotemporal structure of syndrome data. In particular, we present a vectorised method for surface code data simulation and benchmark decoding performance when such data defines a multi-label classification problem and generative modelling problem for rotated surface codes with circuit noise after each gate and idle timestep. Performance was found to generalise to rotated surface codes of sizes up to $d=97$, with depolarisation parameter thresholds of up to $0.7\%$ achieved, competitive with h Minimum Weight Perfect Matching (MWPM). Improved latencies, compared with MWPM alone, were found starting at code distances of $d=33$ and $d=89$ under noise models above and below threshold respectively. These results suggest promising prospects for ANN-based frameworks for surface code decoding with performance sufficient to support the demands expected from fault-tolerant resource estimates.

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

3 major / 2 minor

Summary. The manuscript introduces a fully convolutional 3D neural network decoder for rotated surface codes subject to circuit-level syndrome noise. It reports that the decoder generalizes to distances up to d=97, attaining depolarizing thresholds as high as 0.7% that are competitive with minimum-weight perfect matching (MWPM), together with latency improvements relative to MWPM that begin at d=33 (above threshold) and d=89 (below threshold).

Significance. If the generalization and performance claims are substantiated, the work would constitute a meaningful step toward scalable machine-learning decoders for surface-code quantum error correction. The exploitation of spatiotemporal structure via 3D convolutions directly addresses a known limitation of prior ANN decoders, and competitive thresholds plus latency gains at moderate-to-large distances would be relevant to resource estimates for fault-tolerant architectures.

major comments (3)
  1. [Abstract and Results] Abstract and Results: the central generalization claim to d=97 lacks any reported information on the largest training distance, the number of distinct noise realizations per distance, or scaling ablations (logical error rate versus distance at fixed training size). Without these data it is impossible to determine whether the quoted 0.7% threshold reflects size-independent feature extraction or merely statistical similarity between training and test distributions.
  2. [Methods/Experimental Setup] Methods/Experimental Setup: the manuscript supplies no description of training/validation splits, error-bar methodology, or the precise protocol used to benchmark against MWPM (identical simulated syndromes, same random seeds, identical noise-model parameters). These omissions are load-bearing for the competitiveness and latency claims.
  3. [Results] Results: latency improvements are asserted to begin at d=33 and d=89 under noise models above and below threshold, yet no accompanying plots of latency versus distance, no specification of the exact depolarizing parameter values, and no statistical uncertainties are provided.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'competitive with h Minimum Weight Perfect Matching' contains an apparent typographical artifact and should read 'competitive with Minimum Weight Perfect Matching'.
  2. [Throughout] Notation: consistent spelling ('depolarizing' versus 'depolarisation') and explicit definitions or ranges for all reported threshold values should be supplied.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript on the fully convolutional 3D neural network decoder for surface codes. The comments highlight important areas for improving clarity and substantiating our claims. We address each major comment point by point below and will revise the manuscript to incorporate the requested details where appropriate.

read point-by-point responses

  1. Referee: [Abstract and Results] Abstract and Results: the central generalization claim to d=97 lacks any reported information on the largest training distance, the number of distinct noise realizations per distance, or scaling ablations (logical error rate versus distance at fixed training size). Without these data it is impossible to determine whether the quoted 0.7% threshold reflects size-independent feature extraction or merely statistical similarity between training and test distributions.

    Authors: We agree that these details are essential to support the generalization claims. In the revised manuscript we will explicitly report the largest training distance used, the number of distinct noise realizations generated per distance, and include scaling ablations that plot logical error rate versus distance at fixed training size. These additions will demonstrate that the reported thresholds arise from the model's extraction of size-independent spatiotemporal features rather than distributional overlap between training and test sets. revision: yes

  2. Referee: [Methods/Experimental Setup] Methods/Experimental Setup: the manuscript supplies no description of training/validation splits, error-bar methodology, or the precise protocol used to benchmark against MWPM (identical simulated syndromes, same random seeds, identical noise-model parameters). These omissions are load-bearing for the competitiveness and latency claims.

    Authors: We acknowledge the omission of these methodological details in the current version. The revised manuscript will include a clear description of the training/validation splits, the procedure for computing error bars, and the exact benchmarking protocol against MWPM, confirming that identical simulated syndromes, random seeds, and noise-model parameters were employed for all comparisons. This will strengthen the validity of the competitiveness and latency results. revision: yes

  3. Referee: [Results] Results: latency improvements are asserted to begin at d=33 and d=89 under noise models above and below threshold, yet no accompanying plots of latency versus distance, no specification of the exact depolarizing parameter values, and no statistical uncertainties are provided.

    Authors: We thank the referee for noting this gap in the presentation of results. The revised manuscript will add plots showing latency versus distance for the regimes above and below threshold, specify the precise depolarizing noise parameters at which the comparisons were performed, and report statistical uncertainties derived from the simulation ensemble. These changes will make the latency improvement claims fully transparent and reproducible. revision: yes

Circularity Check

0 steps flagged

Minor self-citation of prior ANN decoding work present but does not force empirical thresholds or generalization claims

full rationale

The paper reports empirical benchmarking results from training a fully convolutional 3D neural network on simulated surface-code syndrome data under circuit noise. Performance claims (generalization to d=97, thresholds up to 0.7%, latency improvements versus MWPM) arise from direct simulation and evaluation rather than any derivation, equation, or fitted parameter that reduces to its own inputs by construction. Self-citations to earlier ANN decoder literature appear in the introduction but are not load-bearing for the reported thresholds or scaling results; the central evidence consists of independent simulation benchmarks. No self-definitional, fitted-input, uniqueness-imported, or ansatz-smuggled steps are present.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard assumptions of supervised learning on simulated data and the existence of exploitable spatiotemporal correlations in syndrome measurements; no new physical entities are postulated.

free parameters (1)
  • neural network weights and architecture hyperparameters
    Trained on simulated syndrome data; specific values not stated in abstract.
axioms (1)
  • domain assumption Syndrome data possesses sufficient spatiotemporal structure to be effectively processed by 3D convolutions for decoding.
    Invoked in the scheme that exploits this structure for multi-label classification and generative modelling.

pith-pipeline@v0.9.0 · 5753 in / 1424 out tokens · 39430 ms · 2026-05-19T08:38:37.828429+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Fast and accurate AI-based pre-decoders for surface codes

    quant-ph 2026-04 unverdicted novelty 7.0

    AI pre-decoders achieve O(1 μs) per round decoding runtimes on GPUs for surface codes while improving logical error rates over global decoding alone and enabling data-driven noise weight estimation.

Reference graph

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