Error-detected surgery on Iceberg codes

Andrea Di Fini; Guido Pupillo; Laura Pecorari; Samuel Crew

REVIEW 2 major objections 4 minor 28 references

Error-detected surgery on Iceberg codes

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Surgery gadgets measure every logical Pauli of an Iceberg code with quadratic error suppression.

desk verdict A solid, clearly written small-code surgery paper that delivers on its explicit promises; the general-N fault-detection claim rests on an unproven imported criterion, but the core small-code results hold up. read the letter →

arxiv 2608.06187 v1 pith:YH2XMSYG submitted 2026-08-06 quant-ph

Andrea Di Fini , Samuel Crew , Laura Pecorari , Guido Pupillo This is my paper

classification quant-ph MSC 81P7081P68

keywords IcebergcodescodesurgerygauginglogicaloperatorsPaulimeasurementfault-detectinggadgetsPauli-basedcomputationautomorphismorbitscircuit-levelsimulation

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

This paper constructs explicit error-detecting surgery gadgets for the high-rate Iceberg codes $[[2N,2N-2,2]]$, giving a way to measure any logical Pauli product without disturbing the other logical qubits. The central assertion is that the gadgets are fault-detecting at the circuit level: every single fault either acts trivially or is caught by a detector and the run is discarded. Circuit-level simulations of every seed gadget of the $[[4,2,2]]$ and $[[6,4,2]]$ codes fit the post-selected logical error rate to $C p^2$, with fitted slopes between $2.00$ and $2.04$. Because the permutation automorphism group of the Iceberg code is the full symmetric group, logical Paulis split into orbits with minimal-weight seeds, so a small number of gadgets---four for $[[4,2,2]]$, five for $[[6,4,2]]$---measures all logical Paulis after a classical relabelling. The paper presents this as a natural near-term experiment on hardware with reconfigurable long-range connectivity, since the small blocks keep acceptance rates practical.

What carries the argument

The carrying object is the surgery gadget attached to a seed operator $P=\prod_{v\in V} P_v$ of weight $w$. One places gauge qubits $\tau_e$ on the edges of a connected auxiliary graph $\Gamma$ over the support, defines Gauss-law checks $G_v = P_v \prod_{e\ni v} \tau_e^X$, and dresses the two Iceberg stabilisers as $W_X = s_X \prod_{e\in D_X} \tau_e^Z$ and $W_Z = s_Z \prod_{e\in D_Z} \tau_e^Z$ with the boundary conditions $\partial D_X = C_X$, $\partial D_Z = C_Z$; cycles of $\Gamma$ contribute flux checks $B_c=\prod_{e\in c}\tau_e^Z$. Cheeger's constant $h(\Gamma)=\min_{0<|S|\le |V|/2} |\partial S|/|S|$ decides distance preservation: the path graph suffices for weight $w\le 3$ seeds, while higher-weight seeds need extra edges---for the weight-four seed $XXZZII$, closing a four-cycle restores $h(\Gamma)=1$. Compilation is carried by the orbit classification under $S_{2N}$, with Algorithm 1 reducing any logical Pauli to a seed and a permutation in $O(N)$ time.

What would settle it

A direct check is to enumerate every single circuit-level fault in the $[[6,4,2]]$ $XXZZII$ gadget and ask whether any fault changes the logical observable while all detectors $D_0$--$D_{18}$ remain silent; one such fault would refute the fault-detecting claim, and none may exist if the fitted slope of $P_{L|\mathrm{acc}}$ is truly quadratic.

Watch

Extended reading notes

Core claim

The central claim is that a logical Pauli operator of an Iceberg code can be measured fault-detectingly by adjoining gauge qubits on a connected auxiliary graph over its support, dressing the two global stabilisers with Wilson lines so the dressed checks commute with the Gauss-law checks, and running a merge--split protocol whose accepted outcomes reproduce the eigenvalue as the product of the Gauss-law outcomes. The orbit classification is a second claim: under the full permutation automorphism group $S_{2N}$, logical Paulis fall into orbits indexed by composition tuples, each with a minimal-weight seed, so one gadget per seed measures every logical Pauli after a free relabelling. The numerical claim, verified by circuit-level simulation for every seed gadget of the $[[4,2,2]]$ and $[[6,4,2]]$ codes, is that the post-selected logical error rate is $\simeq C p^2$ with fitted slopes $2.00$--$2.04$, matching the fault-detecting requirement that every single fault be trivial or detected.

Load-bearing premise

The load-bearing premise is the imported distance-preservation criterion: for a connected auxiliary graph $\Gamma$ over the seed's support, the merged code preserves distance whenever the Cheeger constant satisfies $h(\Gamma) \ge 1$; the paper cites this to [5] rather than proving it, and if it fails a single fault could become an undetected logical error.

Editorial extensions

If this is right

Every logical Pauli of an Iceberg block reduces to one of a small set of seed gadgets in $O(N)$ time; the permutation returned by the reduction is absorbed as a classical change of addressing.
The post-selected logical error rate $\simeq C p^2$ makes each gadget an error-detecting primitive for Pauli-based computation, so a compiled logical circuit becomes a sequence of such measurements with classical frame corrections.
The GHZ example shows a complete small computation on one $[[6,4,2]]$ block using three of the five gadgets and at most four gauge qubits.
Two blocks can execute a joint inter-block Pauli measurement by bridging their gadgets with one gauge qubit and closing a four-cycle, preserving distance two.
The gadget count grows only polynomially ($O(N^3)$ seeds) while the number of logical Paulis grows exponentially, so calibration and compilation stay manageable as the block grows.

Reading between the lines

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

The paper restricts numerical checks to $N=2$ and $N=3$; by the same counting one expects the orbit reduction to hold for all $N$, but the reported acceptance rates decline with gadget size, suggesting the practical demonstration window is likely at the smallest blocks.
Because merge--split is presented as gauge fixing of a subsystem code, the same primitive could in principle be composed with magic-state teleportation to produce error-detected non-Clifford operations; the paper does not demonstrate this.
The Cheeger criterion is imported as a sufficient condition; one could test whether weaker graph conditions preserve distance for Iceberg codes and reduce gauge-qubit overhead for weight greater than four operators.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know this paper delivers exactly what it promises: explicit error-detecting surgery gadgets for the [[4,2,2]] and [[6,4,2]] Iceberg codes, a full orbit classification of logical Paulis under S_2N, a linear-time seed-reduction algorithm, and clean circuit-level simulations showing post-selected logical error rates scaling as p^2 (slopes 2.00–2.04 across all seeds and both Lagrangian runs). The construction is a concrete instantiation of Williamson–Yoder's gauging framework, not a new paradigm, but the concreteness is valuable: someone building a near-term neutral-atom demonstration can take these gadgets and go. The orbit-count formula (Eq. 16) and the seed-reduction algorithm appear to be new and are self-contained. The paper is also honest about its limits: distance-two, post-selection only, no asymptotic fault-tolerance improvement.

The main soft spot is the general-N construction. Section 4.1 asserts that the merged code preserves distance whenever the auxiliary graph has Cheeger constant h(Γ)≥1, citing [5]. That criterion is not proved here, and the paper only verifies N=2,3 where the single-fault enumeration covers all cases directly. If the imported theorem from [5] does not apply to the dressed stabilizer group with Wilson-line dressings and flux checks, the large-N recipe could produce an undetected weight-one logical error. This is a real gap, but it is a gap in the generalization, not in the demonstrated small-code results. I would ask the authors to either prove the criterion for their dressed group or brute-force check the distance for N=4,5. It is also worth noting that no code or data artifacts are shipped, and the flagged-extraction circuits are only cited, so the single-fault enumeration is not independently reproducible from the paper alone. That is a minor complaint for a theoretical paper with simulations, but it matters if someone wants to reuse the gadgets.

Who is this for: experimentalists working on error-detected computation with Iceberg codes, and anyone who wants a worked, self-contained example of surgery-as-gauging. The paper deserves peer review; I would send it to a referee with a request to address the distance-preservation step. A conditional accept seems right.

Referee Report

2 major / 4 minor

Summary. This paper constructs explicit error-detecting surgery gadgets for the [[2N,2N−2,2]] Iceberg codes, building on the gauging-based perspective on code surgery due to Williamson and Yoder. The authors classify logical Pauli operators of the Iceberg code under the full permutation group S_{2N} into O(N^3) orbits, construct one gadget per orbit seed, and provide a complete merge–split measurement protocol with post-selection. They report circuit-level simulations for all seed gadgets of the [[4,2,2]] and [[6,4,2]] codes, finding post-selected logical error rates that scale as C p^2 with fitted slopes between 2.00 and 2.04, and acceptance rates consistent with gadget size. The paper also presents a worked Pauli-based computation example on a single [[6,4,2]] block and develops the gauging/surgery framework pedagogically.

Significance. The central construction is concrete and clean: the orbit classification is proved in Appendix A, the gadget stabiliser tables are explicit, and the numerical slopes provide compelling evidence for the fault-detecting property in the two smallest codes. The paper's emphasis on explicit small gadgets and its pedagogical structure make it a useful reference for near-term experimental demonstrations of surgery on high-rate codes. The use of two Lagrangian completions to certify protection of all logical directions is a nice methodological addition, and the simulation methodology (Stim, with sample sizes adjusted to a 2.5% relative standard error) is credible. If the general-N fault-detection claim is fully supported, the paper provides a polynomial-size gadget toolkit for a broad family of high-rate codes.

major comments (2)

[Section 4.1, Eqs. (21)-(22)] The claim that the merged code preserves the distance whenever h(Γ) ≥ 1 is imported from reference [5] without stating the theorem or verifying that the dressed stabiliser group S_merge of Eq. (21) satisfies its hypotheses. This criterion is load-bearing for the central fault-detection claim of Section 4.2, because a weight-one logical operator in S_merge would turn a single circuit-level fault into an undetected logical error and break the O(p^2) post-selected scaling. The simulation evidence covers only the [[4,2,2]] and [[6,4,2]] gadgets, where the direct single-fault enumeration supports the claim, so the general-N construction is not independently supported within the manuscript. The authors should either provide a proof or a precise statement of the distance-preservation theorem in this setting, or explicitly restrict the fault-detection claim to the simulated codes and present the general-N construction as conjectural.
[Section 4.3] The statement "we also enumerate all single circuit-level faults and confirm that each is either detected or acts trivially" is not accompanied by any details of the enumeration (fault locations, circuit model, number of faults, or code), so the direct check of the fault-detecting property is not reproducible. Please provide the enumeration method, a summary of results, or make the verification code available as supplementary material.

minor comments (4)

[Appendix B, Eq. (85)] The notation τ^x_e is inconsistent with τ^X_e used throughout the rest of the paper; please unify the notation.
[Section 5] There are two typos: "ciruit level" should be "circuit level", and in Section 4.4 the phrase "five[[6,4,2]]gadgets" is missing a space.
[References] The paper relies on reference [5] for the distance-preservation criterion and on [22,26] for flagged syndrome extraction; please include arXiv identifiers or DOIs so that readers can verify these imported results, especially [5].
[Section 4.1] The sentence "The merged code preserves the distance of the original whenever h(Γ) ≥ 1" could be misread as referring to the original code's distance; since the original distance is 2, it would be clearer to write "the merged code has distance at least 2".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central fault-detection claim is verified by independent circuit-level simulation; imported Cheeger criterion and flagged extraction are external, not self-cited.

full rationale

Score 0. I find no circular step in the derivation chain. The central claim is that the constructed gadgets are fault-detecting, i.e. every single circuit-level fault is either trivial or rejected (Section 4.2), so the post-selected logical error rate is O(p^2). This is not assumed: the protocol's detectors (D0-D18) are comparisons of repeated measurements, and the O(p^2) behavior is verified by independent circuit-level Stim simulations (Section 4.3), including a direct enumeration of all single faults and two mutually non-commuting Lagrangian completions to certify every logical direction. No fitted parameter is used to produce the quadratic prediction; the slopes in Table 3 are fitted to the simulation output as a check, not as an input to the construction. The load-bearing imported ingredients are the Cheeger-constant distance-preservation criterion h(Gamma)>=1 (Eq. 22) attributed to [5] and the flagged syndrome extraction attributed to [22,26]. Both are external works by other authors, not self-citations of the present paper; they are parameter-free sufficient conditions stated with assumptions that do not include the target result (no weight-one logical in the merged code). Under the review rules, such external citations are independent evidence and do not raise the circularity score. The orbit classification (Section 3.3, Appendix A) is proved in the paper by a bijection with V4-orbits and Burnside counting, not assumed from the conclusion. I therefore set score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central construction relies on four imported domain assumptions: the Cheeger distance-preservation criterion from [5], the fault tolerance of standard flagged extraction from [22,26], the free relabelling assumption on reconfigurable hardware, and the independent depolarizing noise model. There are no free parameters fitted to data in the derivation; the noise strength p is swept, not fitted, and the fitted slopes and coefficients in Table 3 are reported outputs, not inputs. No new physical entities are postulated.

assumptions (4)

domain assumption A connected auxiliary graph Gamma with Cheeger constant h(Gamma) >= 1 yields a merged code preserving the distance of the original code.
Imported from Williamson-Yoder [5], cited in Section 4.1 after Eq. (22). Used to justify path graphs for weight <=3 seeds and the four-cycle for the XXZZ seed; not re-derived here.
domain assumption Standard flagged syndrome extraction is sufficient to fault-tolerantly measure the high-weight checks X^{2N}, Z^{2N}, WX and WZ at circuit level.
Used in Section 4.2: 'We therefore measure these checks with a standard flagged syndrome extraction [22, 26]'. The actual flag circuits and their fault-path analysis are not included.
domain assumption Permutation automorphisms of the code can be compiled as a free change of qubit addressing on reconfigurable hardware, with no physical permutation applied.
Used in Sections 3.3 and 4.2.1. This is a hardware-connectivity assumption and relies on the code stabilizers being permutation invariant.
domain assumption Circuit-level noise is modelled as independent depolarizing, reset bit-flip and measurement errors with a single parameter p.
Used in Section 4.3. The fault-detection verification is with respect to this noise model; correlated faults are outside the stated scope.

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Cite this review

Pith. "Pith review of Error-detected surgery on Iceberg codes." pith.science (2026). https://pith.science/paper/YH2XMSYG

@misc{pith2026260806187,
  author       = {Pith},
  title        = {Pith review of: Error-detected surgery on Iceberg codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YH2XMSYG}},
  note         = {Machine review of arXiv:2608.06187}
}

abstract

We construct explicit error-detecting surgery gadgets---small systems of auxiliary qubits and checks---for the high-rate Iceberg codes $[[2N,2N-2,2]]$, to perform fault-detected measurements of logical Pauli products. The construction follows the perspective of surgery as the gauging of a logical operator, regarded as a symmetry of the code. We give a complete classification of logical Pauli operators under the permutation automorphism group of the Iceberg code, reducing the construction to one gadget per orbit, and we verify with circuit-level simulations that the gadgets are fault-detecting, with the expected post-selected logical error rate. The gadgets require reconfigurable long-range connectivity, available on platforms such as neutral-atom arrays, making an error-detected demonstration of Pauli-based computation a natural near-term experiment. The paper doubles as a self-contained introduction to gauging and code surgery, developed alongside a simple worked example.

Figures

Figures reproduced from arXiv: 2608.06187 by the authors.

**Figure 2.** A physical layout of the [[2N, 2N − 2, 2]] Iceberg code qubits. 3 The Iceberg code The Iceberg code [16] has a very high encoding rate (the logical to physical qubit ratio) but distance d = 2, so it can detect but not correct errors. The Iceberg nomenclature, and its role as a low-overhead target for near-term hardware, are due to a trapped-ion demonstration [17], where post-selection on the detection of single faul… view at source ↗

**Figure 3.** The auxiliary graph Γ for the weight-four seed XXZZII of the [[6, 4, 2]] Iceberg code. The six data qubits are drawn as outward spokes of a hexagon, with the support of the seed labelled by Pauli type and the two unsupported qubits in grey. Gauge qubits live on the edges of Γ (blue and green): the path along the hexagon is closed into a four-cycle by a chord. Gauss-law checks Gv sit at the pink vertices and the flux… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The panels depict the stabiliser structure of the merged [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Post-selected logical error rate for the [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Acceptance rate for the [[4, 2, 2]] (top) and [[6, 4, 2]] (bottom) seed gadgets, run A (left) and run B (right). The pure type seeds XX, Y Y, ZZ are accepted most often, followed by XY Z and, in the [[6, 4, 2]] code, the weight-four XXZZ seed. Error bars are smaller th…

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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work page 2026

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work page 2016

[3] [2]

Quantum measurements and gates by code deformation.Journal of Physics A: Mathematical and Theoretical, 42(9):095302, 2009

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work page 2009

[4] [3]

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Dominic Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter. Surface code quantum computing by lattice surgery.New Journal of Physics, 14(12):123011, 2012. 24

work page 2012

[5] [4]

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Alexander Cowtan and Simon Burton. Css code surgery as a universal construction. Quantum, 8:1344, 2024

work page 2024

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work page 2022

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arXiv 2024

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Dolev Bluvstein, Harry Levine, Giulia Semeghini, Tout T Wang, Sepehr Ebadi, Marcin Kalinowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, et al. A quantum processor based on coherent transport of entangled atom arrays.Nature, 604 (7906):451–456, 2022

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