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arxiv: 2606.22194 · v1 · pith:5A2WXLGWnew · submitted 2026-06-20 · 🪐 quant-ph · cond-mat.dis-nn

Machine Learning Optimal Quantum Error Correction Thresholds

Dominik Seip , Luis Colmenarez , Markus Schmitt , Markus M\"uller This is my paper

Pith reviewed 2026-06-26 11:30 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nn
keywords quantum error correctioncoherent informationneural network decodersurface codetransformer modelmaximum likelihood decodingpost-selectionnoise threshold
0
0 comments X

The pith

The coherent information sets a sharp lower bound on the binary cross-entropy loss of neural decoders that track logical operators through noisy quantum channels.

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

The paper establishes a direct mathematical connection between the coherent information, which measures the maximum reliable transmission rate through a noisy channel, and the binary cross-entropy loss minimized when training neural network decoders. It develops a transformer architecture trained under maximum likelihood decoding that estimates this quantity for the surface code. The resulting thresholds under code-capacity, phenomenological, and circuit-level noise match known theoretical limits, while the same network outperforms minimum-weight perfect matching as a decoder and supports an optimal soft post-selection procedure based on maximum-likelihood cosets.

Core claim

The coherent information constitutes a sharp lower bound on the achievable loss for any decoder that tracks logical operators across noisy channels, and a transformer network trained to estimate the coherent information via maximum likelihood decoding therefore yields threshold estimates that match theoretical limits while delivering lower logical error rates than minimum-weight perfect matching.

What carries the argument

Transformer-based neural network trained via maximum likelihood decoding to estimate the coherent information, which serves as the lower bound on cross-entropy loss for logical-operator-tracking decoders.

If this is right

  • Threshold estimates for the surface code under code-capacity, phenomenological, and circuit-level noise match known theoretical limits.
  • When deployed as a decoder the network produces lower logical error rates than minimum-weight perfect matching.
  • Soft post-selection that filters on network confidence for each logical operator is optimal and reduces both logical error rate and abort probability.
  • The same maximum-likelihood-coset post-selection strategy remains scalable for larger code distances.

Where Pith is reading between the lines

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

  • The loss-bound connection could be used to certify near-optimality of any decoder whose training loss approaches the coherent information.
  • The approach may generalize to other stabilizer codes once suitable training data for logical operators can be generated.
  • Post-selection optimality proofs based on maximum-likelihood cosets likely extend to any decoder whose output probabilities approximate the coherent-information channel.

Load-bearing premise

The network estimates the coherent information without systematic bias arising from its architecture or the choice of training data.

What would settle it

Numerical results in which the network's predicted thresholds deviate from known theoretical values or in which its logical error rate fails to beat minimum-weight perfect matching under identical surface-code noise models would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.22194 by Dominik Seip, Luis Colmenarez, Markus M\"uller, Markus Schmitt.

Figure 1
Figure 1. Figure 1: FIG. 1: Estimates of the CI obtained using our transformer [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Noisy [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Setup for the CI in the context of QEC. A noiseless [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Circuit for estimating conditional logical probabilities [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Model architecture. Full network overview with circuit-level noise affecting data and ancilla qubits. Stabilizer measure [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Model components. (a) Encoder structure. For depolarizing noise, three layers combine multi-head attention, feedfor [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Decoding performance of our neural network vs. the MWPM decoder. We show (a) code capacity for [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Visualization of (a) standard MLD post-selection [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Decoding performance of our NN decoder with soft post-selection based on output probabilities. (upper panels) Logical [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: The (rotated) surface code [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Illustration of the increasing complexity of syndromes in the training set as a function of the depolarizing noise strength [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Comparison to the exact CI calculation in the [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Estimates of the CI for phenomenological noise, incl. [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Training loss for selected, representative noise rates. We observe that training is most difficult around the threshold. [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Logical error rate vs abort probability. Optimal [PITH_FULL_IMAGE:figures/full_fig_p033_17.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Comparison of the two post-selection schemes, stan [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Illustration of the scalable region of the modified [PITH_FULL_IMAGE:figures/full_fig_p034_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Finite size effects of post-selection. Comparison [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Decoding performance of our NN decoder with soft post-selection based on output probabilities. Complementary to [PITH_FULL_IMAGE:figures/full_fig_p036_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: Circuit of a QEC cycle under depolarizing noise. [PITH_FULL_IMAGE:figures/full_fig_p037_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22: Circuit of a QEC cycle under phenomenological noise. [PITH_FULL_IMAGE:figures/full_fig_p038_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23: Circuit of a QEC cycle under circuit-level noise. [PITH_FULL_IMAGE:figures/full_fig_p039_23.png] view at source ↗
read the original abstract

As quantum computers remain susceptible to noise, QEC is essential for preserving logical information during computations. However, the performance of QEC codes breaks down beyond certain noise thresholds, revealing fundamental limits on their ability to protect quantum information. These limits can be characterized using information-theoretic measures such as the coherent information, which quantifies the maximum rate at which logical information can be reliably transmitted through a noisy quantum channel. In this work, we establish a direct connection between the CI and the binary cross-entropy loss used when training neural network decoders. Specifically, we show that the CI constitutes a sharp lower bound on the achievable loss for decoders that track logical operators across noisy channels. To this end, we develop a transformer-based neural network model based on maximum likelihood decoding. We train this network to estimate the CI and evaluate its performance on the surface code under three noise models: code capacity, phenomenological, and circuit-level noise. Our results demonstrate that the network accurately predicts CI and yields threshold estimates that closely match known theoretical limits. When used as a decoder, the network significantly outperforms the minimum weight perfect matching decoder in terms of logical error rate. We also introduce a novel soft post-selection scheme that independently treats uncertainty in both logical operators and relies on confidence-based filtering of the network's output. We prove that such post-selection strategies, based on the MLD cosets, are optimal, and demonstrate their scalability in terms of both logical error rate and abort probability. These findings establish transformer-based architectures as powerful tools for QEC and provide the first numerical evidence supporting the optimality and scalability of MLD-based post-selection.

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

1 major / 0 minor

Summary. The manuscript claims a direct link between the coherent information (CI) and binary cross-entropy loss for neural-network decoders that track logical operators across noisy channels, proving that CI is a sharp lower bound on achievable loss. A transformer model trained via maximum-likelihood decoding is used to estimate CI for the surface code under code-capacity, phenomenological, and circuit-level noise; the authors report that the network accurately predicts CI, produces threshold estimates matching known theoretical limits, outperforms minimum-weight perfect matching as a decoder, and that a novel soft post-selection scheme based on MLD cosets is optimal and scalable.

Significance. If the central claims hold without systematic bias in the CI estimates, the work would provide a theoretically grounded method for using ML to estimate QEC thresholds and improve decoding performance, with the optimality proof for MLD-based post-selection constituting a clear contribution. The reported numerical agreement with independently known thresholds supplies external grounding.

major comments (1)
  1. [Abstract] Abstract (and results on surface-code evaluations): the claim that the network 'accurately predicts CI' and yields thresholds that 'closely match known theoretical limits' rests on the premise that the transformer trained via maximum-likelihood decoding produces unbiased CI estimates. Any architecture- or data-induced systematic offset would invalidate both the threshold-matching result and the asserted superiority over MWPM, since those conclusions are derived from the lower-bound relation alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this key point about potential bias in the CI estimates. We address it directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and results on surface-code evaluations): the claim that the network 'accurately predicts CI' and yields thresholds that 'closely match known theoretical limits' rests on the premise that the transformer trained via maximum-likelihood decoding produces unbiased CI estimates. Any architecture- or data-induced systematic offset would invalidate both the threshold-matching result and the asserted superiority over MWPM, since those conclusions are derived from the lower-bound relation alone.

    Authors: We agree that the absence of systematic bias is essential for the validity of the threshold and decoder-performance claims. Our central theorem establishes that binary cross-entropy is bounded from below by the coherent information, with equality achieved precisely under maximum-likelihood decoding. The transformer is trained to minimize this loss, which corresponds to MLD. To verify lack of architecture- or data-induced offset, we have performed additional checks on small surface-code instances (distance 3–5) where the exact CI can be computed by enumeration; the network estimates agree with these exact values to within statistical error. In the revised manuscript we will add a dedicated validation subsection presenting these comparisons, together with an analysis of convergence to the MLD limit as a function of training data volume and model capacity. We will also revise the abstract to state that the estimates are consistent with exact results on small instances and match known thresholds within reported uncertainties, thereby making the supporting evidence explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; central claims grounded by independent theoretical thresholds

full rationale

The paper derives that coherent information (CI) is a sharp lower bound on binary cross-entropy loss for decoders tracking logical operators, then trains a transformer via maximum-likelihood decoding to estimate CI on surface-code instances. Threshold estimates are validated by direct numerical agreement with independently known theoretical limits from prior literature on code-capacity, phenomenological, and circuit-level noise; this external match supplies non-circular grounding. The optimality proof for MLD-coset post-selection is self-contained and does not rely on fitted values or self-citations. No step reduces a reported prediction to its own training inputs by construction, and no load-bearing premise collapses to a self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the neural-network architecture and training procedure implicitly introduce hyperparameters whose values are not reported.

pith-pipeline@v0.9.1-grok · 5825 in / 1224 out tokens · 30830 ms · 2026-06-26T11:30:38.937228+00:00 · methodology

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