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REVIEW 2 major objections 5 minor 58 references

A partition function framework for estimating logical error curves in stabilizer codes

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A ratio of partition functions exactly measures the success rate of maximum-likelihood decoding in stabilizer codes, letting logical error curves be estimated from far fewer samples than by counting decoder failures.

desk verdict The decoding-probability / order-probability distinction is real and worth stealing; a stickier-than-stated tie-breaking ambiguity in the MP definition should be fixed but doesn't sink the paper. read the letter →

arxiv 2505.15758 v3 pith:LZOS6MLW submitted 2025-05-21 quant-ph

Leon Wichette , Hans Hohenfeld , Elie Mounzer , Linnea Grans-Samuelsson This is my paper
classification quant-ph MSC 81P7082B2082B44 PACS 03.67.Pp
keywords stabilizercodespartitionfunctiondecodingNishimoritemperaturemaximumlikelihoodorderprobabilitylogicalerrorcurvesensembling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that two ratios of partition functions from the statistical-mechanics mapping of stabilizer codes measure the success rates of two distinct decoding strategies. The decoding probability at temperature $T$ equals the success rate of a maximum partition function decoder, which at the Nishimori temperature is maximum-likelihood (ML) decoding and at zero temperature is degeneracy-enhanced maximum-probability (dMP) decoding. The order probability, by contrast, equals the success rate of a probabilistic partition function decoder, which at zero temperature is ordinary maximum-probability (MP) decoding. This corrects the earlier use of the order probability as a proxy for optimal decoding, and it provides a sample-efficient estimator of logical error curves: roughly 75% fewer samples achieve the same confidence interval as counting decoder failures. The framework is demonstrated on the toric code under uniform and non-uniform bitflip noise.

What carries the argument

The load-bearing object is the ratio of partition functions, $Z_T(C^*_{\vec{s}}(T))/\sum_{C} Z_T(C_{\vec{s}})$, evaluated on the disordered statistical-mechanics model associated to the stabilizer code. For each syndrome, each logical equivalence class carries a partition function, and the Nishimori temperature $T_{\rm Nish}$ (the temperature set by the physical error rates so that partition functions equal class probabilities) is where a maximum partition function decoder becomes a maximum-likelihood decoder. The decoding probability averages, over all error realizations, the weight at $T_{\rm Nish}$ of the class that maximizes $Z_T$ at temperature $T$; the order probability averages the weight at $T$ of the class containing the actual error. At $T=0$ the first counts classes by their number of most-probable errors (dMP) while the second samples errors proportional to that count (MP). Numerically the framework uses the FKT/Pfaffian exact computation of Random Bond Ising Model partition functions for the toric code, with Wang-Landau sampling as a zero-temperature cross-check.

What would settle it

Compare the decoding probability at $T_{\rm Nish}$ against the actual success rate of a brute-force ML decoder on a small toric code (say distance 3 or 4) by enumerating all errors and syndromes exactly; any systematic disagreement between the two curves beyond statistical error would falsify Proposition 2. A less direct test is to search a model with a reentrant phase boundary for a vertical maximum-partition-function decodability boundary, which Corollary 1 says is possible but the toric code example does not exhibit.

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Extended reading notes

Core claim

If correct, the paper's central discovery is that the success rate of a maximum partition function decoder at temperature $T$ is exactly the decoding probability at $T$ (Proposition 2), while the success rate of a probabilistic partition function decoder at $T$ is exactly the order probability at $T$ (Proposition 3). At the Nishimori temperature, where error-class probabilities equal partition functions, the maximum partition function decoder is the ML decoder, so the decoding probability is the exact logical success rate of ML decoding. At zero temperature the two decoders reduce respectively to dMP and plain MP decoding (Proposition 1). A consequence is that the order probability, previously used to locate thresholds, actually measures a suboptimal probabilistic decoder, and the gap between the two probabilities quantifies how much accounting for degeneracy among maximum-probability errors improves a decoder.

Load-bearing premise

The central claim depends on the statistical-mechanics mapping that makes an error class's probability equal to its partition function at the Nishimori temperature; if that mapping fails for a code or noise model, the decoding probability is no longer the ML success rate.

Editorial extensions

If this is right

  • Logical error curves for ML decoding in the toric code can be read directly from decoding-probability samples at $T_{\rm Nish}$, with roughly 75% fewer samples than failure counting for the same confidence interval.
  • The order probability does not measure optimal decoding; it measures a probabilistic partition function decoder that at zero temperature is MP decoding, so earlier threshold estimates based on order probability are thresholds of a suboptimal decoder.
  • At zero temperature the gap between decoding probability and order probability is exactly the gain from dMP over MP decoding, that is, the gain from resolving degeneracy among maximum-probability errors; weak ensembling approximates dMP and inherits this gain.
  • In the toric code under bitflip noise the maximum partition function decodability boundary is also reentrant, and a maximum-Z decoder fails to stay optimal away from the Nishimori line; the paper's Corollary 1 gives conditions under which a decodability boundary could be vertical even with a reentrant phase boundary.
  • Under non-uniform bitflip noise the ground-state degeneracy is lifted so dMP equals MP, yet ML still outperforms both, and ensembling remains useful only when its perturbations are strong enough to sample less probable errors.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Outside this paper: the ratio framework should transfer beyond Pfaffian-computable models, so tensor-network partition-function approximants could yield ML logical error curves for surface codes under circuit-level noise, where no exact ML decoder is available.
  • The sample-efficiency gain grows as the failure rate drops, so decoding-probability estimation may make low-noise logical error rates accessible that would require impractically many Monte Carlo samples by failure counting; a testable prediction is a variance comparison at $p$ well below threshold.
  • The distinction between decodability and phase boundary suggests a search over statistical-mechanics models with reentrant phase boundaries for one where the maximum-partition-function decodability boundary is vertical, using the condition in Corollary 1 as the numerical check.
  • The parity-dependent bias of MWPM decoders quantified here implies that benchmark comparisons between fast decoders and ML should be performed on even-distance codes, where degeneracy and bias effects are strongest.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces two partition-function-based estimators for stabilizer-code logical error rates: the "decoding probability" and the "order probability." It proves that these estimators exactly equal the success rates of two families of decoders: maximum partition function decoders and probabilistic partition function decoders. At the Nishimori temperature the maximum partition function decoder is maximum likelihood (ML), while at zero temperature the two families are respectively identified with degeneracy-enhanced maximum probability (dMP) decoding and maximum probability (MP) decoding. The framework is applied numerically to the toric code under bitflip noise using FKT and Wang-Landau partition function computations, with cross-validation between methods, a demonstration of roughly 75% sample savings over failure counting, and an analysis of MWPM bias, ensembling, parity effects, and non-uniform noise.

Significance. If correct, the paper provides a unified and decoder-independent route to full logical error curves rather than only thresholds, and it clarifies that the order probability used in earlier work is not the ML success rate but rather the success rate of a suboptimal probabilistic decoder. The main propositions are simple and their proofs are transparent, and the numerical section is carefully cross-validated: FKT at T=0.1TNish is checked against Wang-Landau at T=0, and ratio-based estimates are checked against direct decoder-failure counting within error bars. The public availability of the code is a further strength. The principal caveat is a definitional ambiguity about tie-breaking in the MP and dMP identifications, which is load-bearing for the advertised interpretation of the zero-temperature results.

major comments (2)
  1. [Section II A, Eqs. (10)-(11); Proposition 1.3, Eq. (23)] The claim that the order probability at T=0 measures the success rate of MP decoding is only true for an MP decoder that selects a maximum-probability error uniformly at random among all such errors and then returns the logical class of that selected error. As written, Eq. (10) defines C_s^MP as an argmax set, and Eq. (11) is not well defined when more than one class attains the same maximum individual-error probability; different tie-breaking rules give different success rates. Proposition 1.3 in effect introduces the uniform tie-breaking rule, but that rule is not part of the MP definition in Section II A. Please amend the definition and Eq. (11), and qualify the abstract and Section III statements accordingly.
  2. [Section II A, Eq. (12); Definition 1; Proposition 1.2] A closely related issue affects the dMP identification. Eq. (12) defines C_s^dMP as an argmax over classes of n_max(C_s,L|s), while Definition 1 at T=0 chooses uniformly among all classes in {C_s^max(0)}. If multiple classes tie for the largest n_max, these two prescriptions are not the same decoder unless dMP is also defined with uniform tie-breaking. Please state explicitly whether Proposition 1.2 refers to an arbitrary tie-breaking dMP decoder or to the particular uniformly randomized version used in Definition 1, and align Eq. (12)-(13) with that convention.
minor comments (5)
  1. [Acknowledgments] The acknowledgments contain a typo: "We than Steven Simon" should be "We thank Steven Simon."
  2. [References [41] and [45]] Reference [41] and reference [45] appear to be the same paper by Stace and Barrett; please merge the duplicate.
  3. [Fig. 10 caption] The word "ensemling" in the caption of Fig. 10 should be "ensembling."
  4. [Appendix D] In Appendix D the phrase "minumum-weight perfect matching" should be "minimum-weight perfect matching."
  5. [Section V A, Fig. 7] The discussion of weak ensembling bias is clear, but the text would benefit from an explicit statement that the uniform sampling assumed by the order-probability benchmark is not achieved by the weak-ensembling procedure; currently this point is made only indirectly through Fig. 7.

Circularity Check

1 steps flagged · score 1.0 of 10

Minor definitional circularity: the T=0 order-probability/MP identification requires an unstated uniform tie-break; the rest of the framework is self-contained.

  1. self definitional [Section III A, Proposition 1.3 and proof; cf. Section II A, Eqs. (10)-(11)]
    "C^MP_s = arg max_L max_{e in C_s,L} P(e) (10) ... P^MP_success = sum_s P(s)P(C^MP_s | s) (11). ... a probabilistic partition function decoder at zero temperature returns a correction belonging to a class ~C_s with probability P_decoder(~C_s)= n_max(~C_s)/sum_C n_max(C), which is equivalent to returning a maximum probability error e, with equal probability for each such error."

    Eq. (23) is simply Definition 2 at T=0; it is not derived from Section II's MP definition. The earlier definition selects 'the logical class C^MP_s = arg max_L max_{e in C_s,L} P(e)' and Eq. (11) defines the success probability as P(C^MP_s | s) for that single class. When several classes share the maximum error probability, the probabilistic decoder instead returns each class with probability proportional to n_max(C), which is uniform over individual maximum-probability errors. That is a different, randomized tie-breaking convention. The proposition's 'equivalent to returning a maximum probability error e, with equal probability for each such error' supplies the missing convention, making the equality to the order probability true by construction rather than as a consequence of Eq. (11).

full rationale

The rest of the paper is not circular. Proposition 2 and Proposition 3 are transparent identities: the decoders in Definitions 1-2 and the estimators in Definitions 3-4 are built from the same {C_max_s(T)} sets and the same partition-function ratios, so the equalities follow by unpacking definitions rather than by fitting. The statistical-mechanics identification Z_TNish(C)=P(C), Eqs. (14)-(18), is imported from [1,2] as an external mapping, and the numerical logical-error curves are cross-checked by two independent partition-function implementations (FKT at T=0.1 T_Nish versus Wang-Landau at T=0) and against external MWPM/PyMatching benchmarks. No parameter is fitted to produce the decoding-probability or order-probability curves themselves; the ensembling perturbation widths are optimized for a separate matching-decoder comparison and do not enter the central identities. The one genuine caveat is the tie-break ambiguity in Proposition 1.3: identifying the order probability with MP success requires an unstated uniform-over-maximum-errors tie-break, so that particular claim is partly definitional. Since this is a localized, repairable convention issue rather than a hidden input driving the numerical claims, the overall circularity is minor.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central identities are definitional given the standard mapping P(C)=Z_TNish(C). The paper's numerical claims rest on two additional pillars: the low-temperature approximation T=0.1 T_Nish for FKT, and the optimized ensembling widths sigma_xi and sigma_Xi. The noise models (uniform and truncated Gaussian bitflip) are explicit domain assumptions. No free parameters are fitted to produce the reported logical error curves themselves; the fitted quantities concern the ensembling heuristic.

free parameters (3)
  • sigma_xi (ensembling perturbation width, uniform noise) = 10^-6
    Optimized in Appendix E on toric code d=8,16 at p in {0.05,0.07,0.10,0.11}; used for weak ensembling MWPM comparisons in Section V.
  • sigma_Xi (ensembling perturbation width, non-uniform noise) = 0.025
    Optimized in Appendix E across sigma_p in {0.005,...,0.15} for d=8,16; used for strong ensembling in Section VI B.
  • T_approx (finite-temperature stand-in for zero temperature) = 0.1 T_Nish
    Chosen as finite-temperature stand-in for T=0 in FKT partition function computations; validated against WL at T=0 in Fig. 2 but is a hand-chosen approximation.
assumptions (6)
  • domain assumption Error class probabilities equal partition functions at the Nishimori temperature (Eqs. 14-18)
    Adopted from Dennis et al. [1] and Chubb-Flammia [2]; the whole framework depends on this mapping holding for the noise models considered.
  • domain assumption The physical noise is a product of independent single-qubit Pauli errors with known probabilities pi(gi)
    Introduced in Section II A; the non-uniform truncated Gaussian model (Section VI) is an additional modeling assumption.
  • standard math The Pfaffian method correctly gives the RBIM partition function on the torus via 2Z = Pf(K++)+Pf(K+-)+Pf(K-+)+Pf(K--)
    Standard result for planar Ising models (Appendix B), used for all toric code numerics.
  • standard math Wang-Landau density-of-states estimates converge to the true density of states with the chosen flatness and update-factor parameters
    Used as cross-check for zero-temperature limits (Appendix C); its convergence is assumed within stated tolerances.
  • domain assumption T=0.1 T_Nish with 4096 bits of precision approximates the zero temperature limit for FKT
    Validated by comparison with T=0.01 T_Nish (Fig. 14) and WL at T=0 (Figs. 2, 15), but remains an approximation affecting all 'zero-temperature' estimates.
  • domain assumption Weak ensembling with sigma_xi=10^-6 samples only among minimum-weight matchings
    Assumed in Section V A to approximate dMP decoding; the paper notes rare large perturbations can sample higher-weight matchings.

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Pith. "Pith review of A partition function framework for estimating logical error curves in stabilizer codes." pith.science (2026). https://pith.science/paper/LZOS6MLW

@misc{pith2026250515758,
  author       = {Pith},
  title        = {Pith review of: A partition function framework for estimating logical error curves in stabilizer codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LZOS6MLW}},
  note         = {Machine review of arXiv:2505.15758}
}
read the original abstract

Based on the mapping between stabilizer quantum error correcting codes and disordered statistical mechanics models, we define a ratio of partition functions that measures the success probability for maximum partition function decoding, which at the Nishimori temperature corresponds to maximum likelihood (ML) decoding. We show that this ratio differs from the similarly defined order probability and describe the decoding strategy whose success rate is described by the order probability. We refer to the latter as a probabilistic partition function decoding and show that it is the strategy that at zero temperature corresponds to maximum probability (MP) decoding. Based on the difference between the two decoders, we discuss the possibility of a maximum partition function decodability boundary outside the order-disorder phase boundary. At zero temperature, the difference between the two ratios measures to what degree MP decoding can be improved by accounting for degeneracy among maximum probability errors, through methods such as ensembling. We consider in detail the example of the toric code under bitflip noise, which maps to the Random Bond Ising Model. We demonstrate that estimation of logical performance through decoding probability and order probability is more sample efficient than estimation by counting failures of the corresponding decoders, especially in the regime of low noise. We consider both uniform noise and noise where qubits are given individual error rates. The latter noise model lifts the degeneracy among maximum probability errors, but we show that ensembling remains useful as long as it also samples less probable errors. We also consider, in less detail, the color code under bitflip and depolarizing noise.

Figures

Figures reproduced from arXiv: 2505.15758 by the authors.

Figure 1
Figure 1. FIG. 1: A sketch of the Random Bond Ising Model phase diagram, adapted from [ [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Matching within error bars of maximum partition function decoder performance estimates [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Performance estimates for both counting and ratio methods for probabilistic and maximum [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Performance estimates for both counting and ratio methods for probabilistic and maximum [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison of dMP (estimated by decoding probability at [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The left plot shows the confidence interval widths for estimates of performance using decoding [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Depicted are five shortest paths between the black vertices, corresponding to minimum weight [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Comparison of potentially biased matching based decoders to their unbiased partition function [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: In Ref. [ [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Example syndrome pattern for the distance 5 toric code with a chain of two possible matchings [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Comparison of MWPM decoding with ensembling using 50 random perturbations on the decoder [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Comparison of MP, dMP, and ML decoding for varying standard deviations of the cut-off [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Improvement of the logical error rate under ensembling with [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Comparison of FKT performance estimates of maximum partition function decoding at [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Comparison of WL performance estimates of maximum partition function decoding at [PITH_FULL_IMAGE:figures/full_fig_p029_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: Relative domain walls of a spin configuration on [PITH_FULL_IMAGE:figures/full_fig_p030_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: Mapping of domain walls to a dimer cover of [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18: The Pauli- [PITH_FULL_IMAGE:figures/full_fig_p033_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19: Optimization results for [PITH_FULL_IMAGE:figures/full_fig_p034_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20: Optimization results for [PITH_FULL_IMAGE:figures/full_fig_p034_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21: Optimization results for [PITH_FULL_IMAGE:figures/full_fig_p035_21.png]

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Pith tools

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