arxiv: 2605.03054 · v1 · submitted 2026-05-04 · 🪐 quant-ph · cs.DM· math.CO

Recognition: 3 theorem links

· Lean Theorem

Closed form logical error rate approximations for surface codes

Andrea Delgado, Daniel Dilley, Ryan Bennink, Shaked Regev

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:54 UTC · model grok-4.3

classification 🪐 quant-ph cs.DMmath.CO

keywords surface codelogical error ratequantum error correctionerror rate approximationsymmetry countingmeasurement errors

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The pith

Surface codes admit closed-form logical error rate approximations via symmetry-based configuration counting.

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

The paper shows how to calculate logical error rates for surface codes by counting the error configurations that produce a logical failure, using the symmetry of the code. This count yields an approximate closed-form expression for the logical error probability as a function of the physical error rate. The method works under the assumption of independent errors and incorporates measurement errors for more complete comparisons. It enables analysis of different code configurations without running costly classical decoder simulations for each case.

Core claim

Logical error rates in surface codes can be approximated by enumerating the symmetric sets of physical errors that result in a logical failure and converting those counts into a probability polynomial.

What carries the argument

Symmetry-exploiting enumeration of minimal logical-error-inducing error configurations

If this is right

Different surface code distances and geometries can be compared by their logical error rates at any physical error rate.
Measurement errors are included in the counting to compare realistic implementations.
Design choices for hypothetical quantum computers can be evaluated more efficiently.
The approximation accuracy improves as the physical error rate decreases.

Where Pith is reading between the lines

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

If the counting can be automated for larger codes, it may replace simulation entirely for design screening.
Extensions to non-independent errors would require adjusting the probability conversion step.
This approach highlights that surface code performance is determined by the number of low-weight logical error paths.

Load-bearing premise

Physical errors are independent and identically distributed across all locations and times.

What would settle it

Direct comparison of the approximated logical error rate against brute-force enumeration of all error configurations for small surface code distances at various physical error rates.

Figures

Figures reproduced from arXiv: 2605.03054 by Andrea Delgado, Daniel Dilley, Ryan Bennink, Shaked Regev.

**Figure 1.** Figure 1: Rotated surface code that encodes a logical qubit using 25 data qubits and 24 syndrome qubits. XL operators must pass through Z stabilizer plaquettes and ZL operators must pass through X stabilizer plaquettes. An X(Z) error on a data qubit anticommutes with the neighboring Z(X)-type stabilizers and flips the parity (i.e., odd or even) of their measurement outcomes. An additional error on a data qubit adjac… view at source ↗

**Figure 2.** Figure 2: The configuration on the left is more likely because it has fewer errors. However, the decoder cannot distinguish these two configurations. So, if the configuration on the right occurs, the decoder will incorrectly identify it as the configuration on the left and apply the wrong correction. This results in a logical error. subsequent d − 1 steps. This estimate is asymptotically tight because, as d grows, a… view at source ↗

**Figure 3.** Figure 3: Approximating the number of paths Npaths as d2 d−1 becomes increasingly tight asymptotically. The second term in Eq. (5) counts the number of ways to arrange errors along an MLLP. This approximation is a union bound because a fixed set of physical error locations may lie on multiple distinct MLLPs, causing such configurations to be counted more than once. As view at source ↗

**Figure 4.** Figure 4: The code agrees with Eq. (4) with A ≈ 2.09 · 10−1 and pth ≈ 7.33 · 10−2 (R2 = 1 − 6.32 · 10−6 ). to A ≈ 4.04 · 10−1 and pth ≈ 6.22 · 10−2 . Our lower bound in Eq. (6) corresponds to A ≈ 4.04·10−1 and pth ≈ 1.24·10−1 . Asymptotically, the upper bound is substantially closer than the lower bound with respect to the actual count. We fit Eq. (2) to Eq. (4) and get A ≈ 1.62 and pth ≈ 2.49 · 10−1 .= pt−ur for th… view at source ↗

**Figure 5.** Figure 5: The correction factor Pde+1/Pde increases with the code distance d and physical error rate p. However, if k ≪ d 2 , most locations of a k + 1-th error will not result in error cancellation. This means 0.5d 2pPde < Pde+1 < d2pPde . In the regime pd2 ⪆ 1, this contradicts the assumption that de errors leading to a logical error are more likely than de + 1 errors doing so [5]. This could result in less favora… view at source ↗

read the original abstract

We propose a novel method to calculate logical error rates in surface codes, assuming independent and identically distributed physical errors. We show how to use our method to analyze hypothetical quantum computers with various configurations and select designs with lower error rates. Currently, this requires expensive classical simulations of quantum decoders for various distances and physical error rates or inaccurate extrapolation from minimal experimental data. Instead, we use the symmetry of the problem to count the configurations that result in a logical error with our novel software. Given a physical error rate, we can deduce the probability of a logical error, to provably good accuracy. We include an analysis of measurement errors to allow a more complete comparison of different surface code implementations.

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.

Desk Editor's Note private letter to a colleague

Symmetry-based counting for surface code logical errors offers a promising analytical alternative to simulations, though validation details are still needed to confirm the accuracy.

read the letter

The punchline is that the authors use symmetry to count logical-error configurations in surface codes and turn those counts into closed-form probability approximations. This could make parameter optimization faster than running repeated simulations. The new part is the symmetry-driven enumeration technique, implemented in their software, which lets them derive expressions without fitting or heavy Monte Carlo. They extend it to include measurement errors for more realistic comparisons of code implementations. That is a solid step for practical use in designing hypothetical quantum hardware. The work is grounded in direct combinatorial counting under the i.i.d. error model, which avoids circularity. If the enumeration is complete, the accuracy claim could hold. The main concern is whether the symmetry reduction still works fully once measurement errors are added, since they can alter the effective graph and potentially miss some configurations. The abstract asserts provable accuracy from counting, but without explicit bounds or side-by-side simulation checks visible here, that needs confirmation in the full paper. The i.i.d. assumption is standard but restricts the scope. This paper is for quantum error correction specialists who want analytical tools for surface code analysis. Readers focused on speeding up design iterations for larger distances would find it relevant. I recommend sending it to peer review. The method is distinct and addresses a real bottleneck, so referees can evaluate the counting completeness and any numerical support.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a novel combinatorial method for approximating logical error rates in surface codes under the assumption of independent and identically distributed physical errors. It uses symmetry properties of the code lattice to count error configurations that lead to logical failures (after decoding), yielding closed-form probability expressions claimed to have provably good accuracy. The approach is extended to include measurement errors for comparing different surface-code implementations and distances without relying on expensive Monte Carlo simulations of decoders.

Significance. If the symmetry-based enumeration proves exhaustive and the accuracy claims are rigorously bounded, the method could offer a computationally efficient alternative to full simulations for estimating logical error rates across code distances and physical error rates. This would be valuable for optimizing hypothetical quantum computer designs. The explicit treatment of measurement errors strengthens the practical relevance compared to models that ignore them.

major comments (3)

[Abstract and §3] Abstract and §3 (Method): The central claim of 'provably good accuracy' from direct combinatorial counting is asserted without any derivation of error bounds, explicit validation against exact enumeration or decoder simulations, or quantitative assessment of truncation effects in the configuration sum. This leaves the accuracy guarantee unsupported.
[§4] §4 (Measurement Errors): The symmetry reduction used for enumeration is stated to handle measurement errors, yet no verification is provided that all failing syndromes are still captured exactly. Measurement errors break the translational invariance of the error graph, so any missed or over-counted classes would directly scale the deduced logical error rate and undermine the closed-form claim.
[§2] §2 (Assumptions): The conversion from configuration counts to probabilities relies on the IID assumption for physical errors; the manuscript does not analyze how deviations from IID (common in real hardware) propagate into the logical error rate approximation or whether the symmetry counting remains valid.

minor comments (2)

[§3] Notation for the enumerated configuration classes and the resulting multinomial probability terms is introduced without a clear summary table or example for small distances, making it difficult to reproduce the counting procedure.
[Abstract] The abstract mentions 'novel software' for counting but provides no pseudocode, complexity analysis, or link to the implementation, which would be needed to assess scalability for larger code distances.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback. We address each major comment below, indicating revisions where appropriate to strengthen the claims and clarify the method's scope.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (Method): The central claim of 'provably good accuracy' from direct combinatorial counting is asserted without any derivation of error bounds, explicit validation against exact enumeration or decoder simulations, or quantitative assessment of truncation effects in the configuration sum. This leaves the accuracy guarantee unsupported.

Authors: We agree that the manuscript does not include an explicit derivation of the truncation error bound. The combinatorial enumeration is exact for all configurations up to the chosen weight cutoff, and the approximation error is precisely the probability mass of all higher-weight configurations. We will add a new subsection in §3 that derives a rigorous upper bound on this remainder term by bounding the number of weight-k+1 and higher configurations (using the total number of possible error locations on the lattice) and summing the resulting geometric-like series. This will make the 'provably good accuracy' claim for p below a threshold explicit and quantitative. We will also include a brief comparison to small-distance exact enumeration to illustrate the bound in practice. revision: yes
Referee: [§4] §4 (Measurement Errors): The symmetry reduction used for enumeration is stated to handle measurement errors, yet no verification is provided that all failing syndromes are still captured exactly. Measurement errors break the translational invariance of the error graph, so any missed or over-counted classes would directly scale the deduced logical error rate and undermine the closed-form claim.

Authors: We acknowledge that measurement errors reduce the symmetry group compared with the pure data-qubit case. Our enumeration nevertheless remains exhaustive because we explicitly augment the lattice with ancilla qubits and enumerate all combined data-plus-measurement error configurations that produce a logical failure after minimum-weight matching; the reduced symmetry is applied only to the remaining equivalent classes. To address the verification gap, we will add a short subsection in §4 that compares the closed-form expressions against brute-force enumeration of all failing syndromes for d=3 and d=5 (both with and without measurement errors), confirming that no classes are missed or double-counted within the enumerated weight range. revision: yes
Referee: [§2] §2 (Assumptions): The conversion from configuration counts to probabilities relies on the IID assumption for physical errors; the manuscript does not analyze how deviations from IID (common in real hardware) propagate into the logical error rate approximation or whether the symmetry counting remains valid.

Authors: The method is derived under the explicit IID assumption stated in §2; each configuration probability is then simply p^w (1-p)^{n-w} where w is the weight. Under non-IID errors the symmetry counting itself remains valid, but the probability assigned to each configuration must be replaced by the product of the individual qubit error probabilities. We will expand the discussion in §2 to note this limitation and to indicate how the same enumerated classes can be re-weighted when per-qubit error rates are known, while emphasizing that the closed-form expressions in the current manuscript apply strictly to the IID case. revision: yes

Circularity Check

0 steps flagged

Direct symmetry-based enumeration of logical-error configurations yields probabilities by explicit summation under i.i.d. model

full rationale

The derivation proceeds by enumerating failing configurations via lattice symmetry, then converting counts to probabilities using the multinomial expansion of the i.i.d. error model. This step is self-contained: the probability of any given configuration is fixed by the physical error rate p and the number of errors in that configuration; no parameter is fitted to the target logical-error rate, no quantity is defined in terms of itself, and no load-bearing premise rests on a self-citation whose validity is presupposed by the present work. Measurement-error analysis is included by extending the same counting procedure to the space-time graph; any incompleteness would be a correctness issue, not a circular reduction of the claimed formula to its inputs. The method therefore supplies an independent combinatorial expression rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the domain assumption of i.i.d. physical errors and the mathematical symmetry of the surface code lattice that permits exhaustive configuration counting.

axioms (1)

domain assumption Physical errors are independent and identically distributed (i.i.d.).
Explicitly stated in the abstract as the basis for deducing probabilities from counts.

pith-pipeline@v0.9.0 · 5412 in / 1050 out tokens · 25339 ms · 2026-05-08T18:54:31.927082+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost (J-cost machinery) Jcost / washburn_uniqueness_aczel unclear
L = sum_{k=0}^{d^2} C_k p^k (1-p)^{d^2-k}; C_{d_e} ≤ d·2^{d-1}·C(d,d_e)
IndisputableMonolith.Foundation.AlphaDerivationExplicit alphaProvenanceCert (parameter-free constants) unclear
L ≈ A(p/p_th)^{d_e} with A ≈ 2.09·10^-1 and p_th ≈ 7.33·10^-2 fit empirically from path counting

Reference graph

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