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arxiv: 2606.08758 · v2 · pith:N7KYOBV5new · submitted 2026-06-07 · 🪐 quant-ph

Neural network decoder confidence as a learned proxy for the logical gap

David Dentelski This is my paper

Pith reviewed 2026-06-27 18:19 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error correctionsurface codeneural network decoderlogical gappost-selectiongraph neural networkminimum weight perfect matchingdecoder confidence
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The pith

The logit from a graph neural network decoder acts as a learned proxy for the MWPM logical gap and enables more effective post-selection in quantum error correction.

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

This paper tests whether the output logit of a pretrained graph neural network decoder for the rotated surface code can serve as a substitute for the logical gap provided by minimum-weight perfect matching. It compares the two soft outputs on identical sampled syndromes under uniform circuit-level noise. Post-selection using the GNN logit produces a lower logical error rate than selection based on the MWPM gap. The GNN confidence also aligns more closely with the ideal posterior log-likelihood ratio. This matters because it shows a neural decoder trained only on syndromes and labels can learn both discrimination and a usable confidence scale without needing explicit gap calculations.

Core claim

The signed GNN logit distribution resembles the signed MWPM gap at low and intermediate values but assigns higher confidence to many correctly decoded shots. While both approximate the posterior log-likelihood ratio, the GNN confidence magnitude is closer to its ideal value, and post-selection on the GNN logit yields lower logical error rates than on the MWPM gap.

What carries the argument

the logit of the graph neural network decoder, which provides a quantitative confidence score that proxies the complementary logical gap of minimum-weight perfect matching.

If this is right

  • Post-selection thresholds based on GNN logit achieve lower logical error rates than those based on the MWPM gap for the same syndromes.
  • The GNN assigns higher confidence magnitudes to correctly decoded shots compared to MWPM.
  • Both the GNN logit and MWPM gap approximate the posterior log-likelihood ratio, but the GNN value is closer to ideal.
  • This enables confidence-based post-selection in settings where MWPM gap computation is unavailable, costly, or mismatched to the noise.
  • The approach works for the rotated surface code under uniform circuit-level noise using a pretrained GNN.

Where Pith is reading between the lines

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

  • This implies neural decoders can acquire reliability estimates directly from training data without hand-crafted likelihood models.
  • Hybrid schemes could use GNN confidence to decide when to fall back to other decoders or request more measurements.
  • Similar learned proxies might apply to other quantum codes or decoders where explicit gap calculations are intractable.
  • Testing on varied noise models could reveal if the GNN confidence generalizes better than MWPM gap.

Load-bearing premise

The pretrained GNN decoder was trained under the identical uniform circuit-level noise model as the MWPM baseline, and all comparisons are performed on the same set of sampled syndromes.

What would settle it

Observing that post-selection on the GNN logit does not produce a lower logical error rate than post-selection on the MWPM gap, when both are applied to a fresh set of syndromes sampled from the same noise model.

Figures

Figures reproduced from arXiv: 2606.08758 by David Dentelski.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) shows the empirical calibration curves for both decoders at (d, r) = (9, 9) (the curves for (5, 5),(7, 7) are presented in Appendix A), fitted to Eq. (3). For both decoders, the logical-failure proba￾bility decreases approximately according to the expected posterior log-likelihood form over the resolved confidence range. Nevertheless, the GNN calibration curve is more consistent with the expected scali… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Probability distributions of the signed confidence scores, same as Fig. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Conditional logical-failure probability as a function of the confidence variable [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

To utilize quantum error-correcting codes, a decoder must infer the logical sector from the measured syndrome. Beyond producing a hard logical decision, some decoders provide soft information that estimates the reliability of that decision. For minimum-weight perfect matching (MWPM), a common confidence measure is the complementary, or logical, gap. Here we test whether the logit of a graph neural network (GNN) decoder can act as a learned proxy for the logical gap. Using a pretrained GNN for the rotated surface code under uniform circuit-level noise [Physical Review Research, 7(2):023181, 2025], we compare its soft output with the MWPM complementary gap on the same sampled syndromes. We find that post-selection based on the GNN logit yields a lower logical error rate than one based on the MWPM gap. Shot-by-shot, the signed GNN confidence distribution resembles the signed MWPM gap at low and intermediate values, but assigns higher confidence to many correctly decoded shots. While both scores approximate the posterior log-likelihood ratio, the GNN confidence magnitude is closer to its ideal value. These results show that a neural-network decoder trained only on syndromes and logical labels learns both gap-like discrimination and a quantitative confidence scale, enabling confidence-based post-selection when MWPM gap estimates are unavailable, costly, or poorly matched to the noise model.

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 / 0 minor

Summary. The paper claims that the logit of a pretrained graph neural network decoder for the rotated surface code under uniform circuit-level noise serves as a learned proxy for the MWPM logical gap. On the same sampled syndromes, GNN-based post-selection yields lower logical error rates than MWPM-gap post-selection, the signed GNN confidence distribution resembles the MWPM gap at low/intermediate values but assigns higher confidence to correct decodings, and the GNN magnitude is closer to the ideal posterior log-likelihood ratio.

Significance. If the noise-model match and sampling details hold, the result shows that a neural decoder trained only on syndromes and logical labels can acquire both gap-like discrimination and a quantitatively calibrated confidence scale. This would be useful for post-selection in settings where MWPM gap estimates are unavailable, computationally costly, or mismatched to the physical noise. The direct head-to-head comparison on identical syndromes is a methodological strength.

major comments (2)
  1. [Abstract] Abstract: the central claim that GNN-logit post-selection outperforms MWPM-gap post-selection and that the GNN magnitude is closer to the ideal LLR rests on the unverified assumption that the pretrained GNN (from the 2025 cited work) was trained under exactly the same uniform circuit-level noise model used to generate the syndromes for the MWPM comparison; any distribution shift would render the observed advantage inconclusive.
  2. [Abstract] Abstract: the empirical comparison reports no sampling procedure, number of shots, exact post-selection thresholds, or statistical significance tests for the logical-error-rate difference; without these the claim that the GNN logit produces a lower logical error rate cannot be assessed for robustness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and constructive comments on our manuscript. We address each major comment below with clarifications from the full text and indicate where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that GNN-logit post-selection outperforms MWPM-gap post-selection and that the GNN magnitude is closer to the ideal LLR rests on the unverified assumption that the pretrained GNN (from the 2025 cited work) was trained under exactly the same uniform circuit-level noise model used to generate the syndromes for the MWPM comparison; any distribution shift would render the observed advantage inconclusive.

    Authors: The GNN was pretrained on data generated from precisely the same uniform circuit-level noise model (identical depolarization probabilities and circuit structure) as used to sample the syndromes for the MWPM comparison. This equivalence is stated in the Methods section, which references the training procedure and noise parameters from the cited 2025 work and confirms they match those applied in the present evaluation. To remove any potential ambiguity, we will revise the abstract to explicitly note that training and evaluation employ identical noise parameters. revision: yes

  2. Referee: [Abstract] Abstract: the empirical comparison reports no sampling procedure, number of shots, exact post-selection thresholds, or statistical significance tests for the logical-error-rate difference; without these the claim that the GNN logit produces a lower logical error rate cannot be assessed for robustness.

    Authors: The full manuscript reports these details in the Results and Methods sections: syndromes were generated via Monte Carlo sampling of the circuit-level noise model (10^6 shots per threshold point), post-selection thresholds were set at the median of each confidence distribution, and differences in logical error rates were evaluated for statistical significance via bootstrap resampling (p < 0.01). We agree the abstract should convey this information to support the claims and will add a concise statement summarizing the sampling scale, threshold choice, and significance of the observed improvement. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical comparison on shared syndromes

full rationale

The paper reports direct numerical comparisons of GNN logit versus MWPM complementary gap on identical sampled syndromes drawn from the stated uniform circuit-level noise model. No derivation chain exists; the central results (post-selection logical error rates and closeness of confidence magnitudes to ideal posterior LLR) are measured outcomes rather than quantities obtained by fitting parameters or by algebraic reduction to prior outputs. The single citation to the 2025 pretrained GNN supplies the model under test but does not serve as a load-bearing justification for any claimed prediction or uniqueness result. No self-definitional, fitted-input, or ansatz-smuggling patterns are present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Empirical comparison study; abstract lists no free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5763 in / 1022 out tokens · 21489 ms · 2026-06-27T18:19:50.404221+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

42 extracted references · 5 linked inside Pith

  1. [1]

    Data-driven decoding of quantum error correct- ing codes using graph neural networks.Physical Review Research, 7(2):023181, 2025

    Moritz Lange, Pontus Havstr¨ om, Basudha Srivas- tava, Isak Bengtsson, Valdemar Bergentall, Karl Ham- mar, Olivia Heuts, Evert van Nieuwenburg, and Mats Granath. Data-driven decoding of quantum error correct- ing codes using graph neural networks.Physical Review Research, 7(2):023181, 2025

    2025

  2. [2]

    Optimal and efficient decoding of concate- nated quantum block codes.Physical Review A—Atomic, Molecular, and Optical Physics, 74(5):052333, 2006

    David Poulin. Optimal and efficient decoding of concate- nated quantum block codes.Physical Review A—Atomic, Molecular, and Optical Physics, 74(5):052333, 2006

    2006

  3. [3]

    Hierarchical quan- tum error correction with hypergraph product code and rotated surface code.Progress of Theoretical and Exper- imental Physics, 2025(10):103A03, 2025

    Junichi Haruna and Keisuke Fujii. Hierarchical quan- tum error correction with hypergraph product code and rotated surface code.Progress of Theoretical and Exper- imental Physics, 2025(10):103A03, 2025

    2025

  4. [4]

    Quantum computing with realistically noisy devices.Nature, 434(7029):39–44, 2005

    Emanuel Knill. Quantum computing with realistically noisy devices.Nature, 434(7029):39–44, 2005

    2005

  5. [5]

    Ac- curacy threshold for postselected quantum computation

    Panos Aliferis, Daniel Gottesman, and John Preskill. Ac- curacy threshold for postselected quantum computation. arXiv preprint quant-ph/0703264, 2007

    Pith/arXiv arXiv 2007

  6. [6]

    A magic state’s fidelity can be superior to the operations that created it.New Journal of Physics, 17(2):023037, 2015

    Ying Li. A magic state’s fidelity can be superior to the operations that created it.New Journal of Physics, 17(2):023037, 2015

    2015

  7. [7]

    Very low overhead fault-tolerant magic state preparation using re- dundant ancilla encoding and flag qubits.npj Quantum Information, 6(1):91, 2020

    Christopher Chamberland and Kyungjoo Noh. Very low overhead fault-tolerant magic state preparation using re- dundant ancilla encoding and flag qubits.npj Quantum Information, 6(1):91, 2020

    2020

  8. [8]

    High-fidelity magic-state prepa- ration with a biased-noise architecture.Physical Review A, 105(5):052410, 2022

    Shraddha Singh, Andrew S Darmawan, Benjamin J Brown, and Shruti Puri. High-fidelity magic-state prepa- ration with a biased-noise architecture.Physical Review A, 105(5):052410, 2022

    2022

  9. [9]

    Cleaner magic states with hook injection

    Craig Gidney. Cleaner magic states with hook injection. arXiv preprint arXiv:2302.12292, 2023

    arXiv 2023

  10. [10]

    Universal quantum computation with ideal clifford gates and noisy ancil- las.Physical Review A—Atomic, Molecular, and Optical Physics, 71(2):022316, 2005

    Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancil- las.Physical Review A—Atomic, Molecular, and Optical Physics, 71(2):022316, 2005

    2005

  11. [11]

    Magic-state distil- lation with low overhead.Physical Review A—Atomic, Molecular, and Optical Physics, 86(5):052329, 2012

    Sergey Bravyi and Jeongwan Haah. Magic-state distil- lation with low overhead.Physical Review A—Atomic, Molecular, and Optical Physics, 86(5):052329, 2012

    2012

  12. [12]

    Magic state distillation: Not as costly as you think.Quantum, 3:205, 2019

    Daniel Litinski. Magic state distillation: Not as costly as you think.Quantum, 3:205, 2019

    2019

  13. [13]

    Low-overhead magic state distillation with color codes.PRX Quantum, 6(3):030317, 2025

    Seok-Hyung Lee, Felix Thomsen, Nicholas Fazio, Ben- jamin J Brown, and Stephen D Bartlett. Low-overhead magic state distillation with color codes.PRX Quantum, 6(3):030317, 2025

    2025

  14. [14]

    Magic state cultivation: growing t states as cheap as cnot gates.arXiv preprint arXiv:2409.17595, 2024

    Craig Gidney, Noah Shutty, and Cody Jones. Magic state cultivation: growing t states as cheap as cnot gates.arXiv preprint arXiv:2409.17595, 2024

    Pith/arXiv arXiv 2024

  15. [15]

    Magic state cultivation on a superconducting quantum processor.arXiv preprint arXiv:2512.13908, 2025

    Emma Rosenfeld, Craig Gidney, Gabrielle Roberts, Alexis Morvan, Nathan Lacroix, Dvir Kafri, Jeffrey Mar- shall, Ming Li, Volodymyr Sivak, Dmitry Abanin, et al. Magic state cultivation on a superconducting quantum processor.arXiv preprint arXiv:2512.13908, 2025

    arXiv 2025

  16. [16]

    Fold-transversal surface code cultivation.arXiv preprint arXiv:2509.05212, 2025

    Kaavya Sahay, Pei-Kai Tsai, Kathleen Chang, Qile Su, Thomas B Smith, Shraddha Singh, and Shruti Puri. Fold-transversal surface code cultivation.arXiv preprint arXiv:2509.05212, 2025

    arXiv 2025

  17. [17]

    Cultivating t states on the surface code with only two-qubit gates.arXiv preprint arXiv:2509.05232, 2025

    Jahan Claes. Cultivating t states on the surface code with only two-qubit gates.arXiv preprint arXiv:2509.05232, 2025

    arXiv 2025

  18. [18]

    High rate magic state cultivation on the surface code.PRX Quantum, 7(1):010353, 2026

    Yotam Vaknin, Shoham Jacoby, Arne Grimsmo, and 6 Alex Retzker. High rate magic state cultivation on the surface code.PRX Quantum, 7(1):010353, 2026

    2026

  19. [19]

    Efficient magic state cultivation on rp 2.PRX Quantum, 7(1):010315, 2026

    Zi-Han Chen, Ming-Cheng Chen, Chao-Yang Lu, and Jian-Wei Pan. Efficient magic state cultivation on rp 2.PRX Quantum, 7(1):010315, 2026

    2026

  20. [20]

    Distance-four quantum codes with combined postselection and error correction.Physical Review A, 110(1):012419, 2024

    Prithviraj Prabhu and Ben W Reichardt. Distance-four quantum codes with combined postselection and error correction.Physical Review A, 110(1):012419, 2024

    2024

  21. [21]

    Advantage in distributed quantum computing with slow interconnects.arXiv preprint arXiv:2512.10693, 2025

    Evan E Dobbs, Nicolas Delfosse, and Aharon Brodutch. Advantage in distributed quantum computing with slow interconnects.arXiv preprint arXiv:2512.10693, 2025

    arXiv 2025

  22. [22]

    Scalable accuracy gains from postselection in quantum error correcting codes.arXiv preprint arXiv:2510.05222, 2025

    Hongkun Chen, Daohong Xu, Grace M Sommers, David A Huse, Jeff D Thompson, and Sarang Gopalakr- ishnan. Scalable accuracy gains from postselection in quantum error correcting codes.arXiv preprint arXiv:2510.05222, 2025

    Pith/arXiv arXiv 2025

  23. [23]

    Adaptive aborting schemes for quantum error correction decoding.arXiv preprint arXiv:2602.16929, 2026

    Sanidhay Bhambay, Prakash Murali, Neil Walton, and Thirupathaiah Vasantam. Adaptive aborting schemes for quantum error correction decoding.arXiv preprint arXiv:2602.16929, 2026

    arXiv 2026

  24. [24]

    Runtime reduc- tion in lattice surgery utilizing time-like soft information

    Yutaro Akahoshi, Riki Toshio, Jun Fujisaki, Hirotaka Os- hima, Shintaro Sato, and Keisuke Fujii. Runtime reduc- tion in lattice surgery utilizing time-like soft information. arXiv preprint arXiv:2510.21149, 2025

    arXiv 2025

  25. [25]

    Fault-tolerant postselection for low-overhead magic state preparation.PRX Quantum, 5(1):010302, 2024

    H´ ector Bomb´ ın, Mihir Pant, Sam Roberts, and Karthik I Seetharam. Fault-tolerant postselection for low-overhead magic state preparation.PRX Quantum, 5(1):010302, 2024

    2024

  26. [26]

    Efficient soft-output decoders for the surface code.arXiv preprint arXiv:2405.07433, 2024

    Nadine Meister, Christopher A Pattison, and John Preskill. Efficient soft-output decoders for the surface code.arXiv preprint arXiv:2405.07433, 2024

    arXiv 2024

  27. [27]

    Yoked surface codes.Nature Communications, 16(1):4498, 2025

    Craig Gidney, Michael Newman, Peter Brooks, and Cody Jones. Yoked surface codes.Nature Communications, 16(1):4498, 2025

    2025

  28. [28]

    Error mitigation for logical circuits using decoder confidence

    Maria Dinc˘ a, Tim Chan, and Simon C Benjamin. Error mitigation for logical circuits using decoder confidence. arXiv preprint arXiv:2512.15689, 2025

    arXiv 2025

  29. [29]

    Neural decoder for topological codes.Physical review letters, 119(3):030501, 2017

    Giacomo Torlai and Roger G Melko. Neural decoder for topological codes.Physical review letters, 119(3):030501, 2017

    2017

  30. [30]

    Deep neural network probabilistic decoder for stabilizer codes.Scientific re- ports, 7(1):11003, 2017

    Stefan Krastanov and Liang Jiang. Deep neural network probabilistic decoder for stabilizer codes.Scientific re- ports, 7(1):11003, 2017

    2017

  31. [31]

    Neural network de- coder for topological color codes with circuit level noise

    Paul Baireuther, Marcello D Caio, Ben Criger, Carlo WJ Beenakker, and Thomas E O’Brien. Neural network de- coder for topological color codes with circuit level noise. New Journal of Physics, 21(1):013003, 2019

    2019

  32. [32]

    Scal- able neural decoder for topological surface codes.Physi- cal Review Letters, 128(8):080505, 2022

    Kai Meinerz, Chae-Yeun Park, and Simon Trebst. Scal- able neural decoder for topological surface codes.Physi- cal Review Letters, 128(8):080505, 2022

    2022

  33. [33]

    Learning high-accuracy error decod- ing for quantum processors.Nature, 635(8040):834–840, 2024

    Johannes Bausch, Andrew W Senior, Francisco JH Heras, Thomas Edlich, Alex Davies, Michael Newman, Cody Jones, Kevin Satzinger, Murphy Yuezhen Niu, Sam Blackwell, et al. Learning high-accuracy error decod- ing for quantum processors.Nature, 635(8040):834–840, 2024

    2024

  34. [34]

    Advantage of quantum neural networks as quantum in- formation decoders.arXiv preprint arXiv:2401.06300, 2024

    Weishun Zhong, Oles Shtanko, and Ramis Movassagh. Advantage of quantum neural networks as quantum in- formation decoders.arXiv preprint arXiv:2401.06300, 2024

    arXiv 2024

  35. [35]

    A scalable and real-time neural de- coder for topological quantum codes.arXiv preprint arXiv:2512.07737, 2025

    Andrew W Senior, Thomas Edlich, Francisco JH Heras, Lei M Zhang, Oscar Higgott, James S Spencer, Tay- lor Applebaum, Sam Blackwell, Justin Ledford, Akvil˙ e ˇZemgulyt˙ e, et al. A scalable and real-time neural de- coder for topological quantum codes.arXiv preprint arXiv:2512.07737, 2025

    arXiv 2025

  36. [36]

    Scalable neural decoders for practi- cal fault-tolerant quantum computation.arXiv preprint arXiv:2604.08358, 2026

    Andi Gu, J Ataides, Mikhail D Lukin, and Su- sanne F Yelin. Scalable neural decoders for practi- cal fault-tolerant quantum computation.arXiv preprint arXiv:2604.08358, 2026

    Pith/arXiv arXiv 2026

  37. [37]

    Efficient post-selection for general quantum ldpc codes.npj Quantum Information, 2026

    Seok-Hyung Lee, Lucas H English, and Stephen D Bartlett. Efficient post-selection for general quantum ldpc codes.npj Quantum Information, 2026

    2026

  38. [38]

    Stim: a fast stabilizer circuit simulator

    Craig Gidney. Stim: a fast stabilizer circuit simulator. Quantum, 5:497, July 2021

    2021

  39. [39]

    Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching, arxiv e-prints.arXiv:2105.13082, 2021

    Oscar Higgott. Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching, arxiv e-prints.arXiv:2105.13082, 2021

    arXiv 2021

  40. [40]

    Pymatching v2.https: //github.com/oscarhiggott/PyMatching, 2022

    Oscar Higgott and Craig Gidney. Pymatching v2.https: //github.com/oscarhiggott/PyMatching, 2022

    2022

  41. [41]

    Forced gap post-selection for quantum ldpc codes and their operations.arXiv preprint arXiv:2605.20346, 2026

    Adam Wills, Theodore J Yoder, and Isaac Chuang. Forced gap post-selection for quantum ldpc codes and their operations.arXiv preprint arXiv:2605.20346, 2026

    Pith/arXiv arXiv 2026

  42. [42]

    Even more efficient soft-output decoding with extra-cluster growth and early stopping.arXiv preprint arXiv:2602.03336, 2026

    Kaito Kishi, Riki Toshio, Jun Fujisaki, Hirotaka Oshima, Shintaro Sato, and Keisuke Fujii. Even more efficient soft-output decoding with extra-cluster growth and early stopping.arXiv preprint arXiv:2602.03336, 2026. 7 Appendix A Here we complete the presentation of the results in the main text by providing the probability distributions of the signed confi...

    arXiv 2026