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REVIEW 3 major objections 6 minor 54 references

X-Z Round Scheduling for the Surface Code with Defects under Biased Noise

T0 review · 3 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Under biased noise, reallocating X- and Z-check rounds in defect-adapted surface codes lowers logical error rates by up to 8.46x.

desk verdict New and plausible scheduling knob for defect-adapted surface codes, but the p_Y=0 model makes the headline LER reductions unreliable. read the letter →

arxiv 2608.05518 v1 pith:QIDSYTVN submitted 2026-08-06 quant-ph

Lakshika Rathi , Pau Escofet , Joshua Viszlai , Margaret Martonosi This is my paper · ORCID
classification quant-ph
keywords surfacecodequantumerrorcorrectionbiasednoisedefectadaptationroundschedulingsyndromeextractionsuper-stabilizerslogicalrate
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 argues that when a surface code is adapted to fabrication defects, the alternating X/Z syndrome-check schedule imposed by the adaptation is not a fixed cost but a tunable design parameter. Under biased noise, where one Pauli error type dominates, measuring the syndrome basis that detects the dominant error more often than the other lowers the logical error rate. In circuit-level simulations with bias $\eta = p_Z/p_X$ up to 10, the optimal ratio $R_X:R_Z$ moves away from 1:1 and improves the average logical error rate by 1.12x to 8.46x. The optimal ratio is set mainly by the noise bias and is nearly independent of code distance and defect rate, so it can be chosen from device calibration data instead of per-device simulation. The same scheduling idea also helps defect-free codes under CNOT crosstalk, where separating X and Z rounds improves the logical error rate by up to 4.5x.

What carries the argument

The central object is the scheduling ratio $R = R_X:R_Z$, the relative number of consecutive syndrome-extraction rounds devoted to X-type versus Z-type gauge checks in a defect-adapted surface code. Defect adaptation converts affected stabilizers into lower-weight gauge checks that anti-commute across bases, so X and Z super-stabilizers must be measured in alternating rounds; the paper treats the relative frequency of those rounds as a free parameter. The mechanism has two effects: more frequent rounds in one basis sample the corresponding syndrome sooner, suppressing the logical error rate in that basis, and repeating the same gauge-check type across consecutive rounds makes individual gauge outcomes deterministic, letting the decoder discount measurement errors. The optimal ratio balances these effects and turns out to be governed by the noise bias $\eta$ rather than by code size or defect density.

What would settle it

Run the same scheduling sweep at $\eta=5$, $d=13$, and 1-2% defect rates with a biased Pauli channel that includes nonzero $p_Y$ (for example $p_Y = \sqrt{p_X p_Z}$), and check whether the optimal ratio $R_X:R_Z$ shifts from the $p_Y=0$ prediction and whether the logical error rate improvement over 1:1 shrinks.

Watch

Extended reading notes

Core claim

The paper's central claim is that under biased noise, the uniform $R=1:1$ round schedule used by defect-adapted surface codes is suboptimal, and reallocating rounds toward the basis that detects the dominant error type lowers the average logical error rate. Concretely, at $\eta=5$ and distance 13, the optimal schedule improves the logical error rate by 4.25x at a 1% defect rate and 8.46x at a 2% defect rate, relative to 1:1. The optimal ratio $R^*$ is, to a good approximation, determined only by the noise bias $\eta$: it stays constant across code distances $d=5$ through 13 and across defect rates 0.5% through 2%, up to sampling variation. This claim extends beyond defective hardware: in defect-free patches where parallel CNOTs suffer crosstalk, measuring X and Z stabilizers in separate rounds reduces the parallel-CNOT count per layer and yields up to a 4.5x logical error rate improvement at $d=13$, $\alpha=2$, and $\eta=5$.

Load-bearing premise

The simulation fixes $p_Y=0$ and assumes Y errors flip both X and Z syndromes, so including them would only rescale the logical error rate at fixed bias rather than change the optimal scheduling ratio; if Y errors interact differently with the alternating gauge-check schedule, the optimal ratio and the reported improvements could shift.

Editorial extensions

If this is right

  • A manufacturer can measure the bias $\eta$ from calibration data and set the scheduling ratio once, reusing it across all code distances without running per-device simulation sweeps.
  • Larger code distances amplify the benefit: at $\eta=5$, the improvement over 1:1 grows from 2.07x at $d=9$ to 4.46x at $d=13$.
  • Higher defect rates increase the benefit: at $d=13$ and $\eta=5$, the gain rises from 4.25x at 1% defects to 8.46x at 2% defects, even though absolute logical error rates worsen.
  • In defect-free architectures with CNOT crosstalk, scheduling X-only and Z-only rounds separately can reduce the logical error rate by up to 4.5x at $d=13$ with a 2x crosstalk penalty.

Reading between the lines

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

  • The same round-rebalancing logic should apply to other defect-adaptation schemes that produce anti-commuting gauge checks, so the ratio could become a universal compilation knob rather than a setting specific to the defect-adaptation framework used here.
  • The $p_Y=0$ assumption is the main point to test on hardware: because Y errors flip both X and Z syndromes, a realistic nonzero $p_Y$ may rescale the optimal ratio or dampen the reported improvements, so a sweep with nonzero Y error would bound the effect.
  • Spatially heterogeneous bias, like the qubit-to-qubit variation shown in the calibration data, suggests a future extension: a position-dependent scheduling ratio that dedicates more rounds to locally dominant error regions rather than a single global optimum.
  • The crosstalk extension implies a trade-off between parallelism and error sampling: separating X and Z rounds halves per-layer CNOT concurrency but doubles the number of rounds, so the 4.5x gain is tied to the simulated crosstalk penalty and could weaken for smaller $\alpha$.
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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

3 major / 6 minor

Summary. This paper studies the optimal ratio of X-type to Z-type syndrome-extraction rounds for a defect-adapted surface code (using the Snakes and Ladders framework) under biased noise. The authors simulate memory experiments with Stim and PyMatching, setting p_Y=0 and fixing p=p_X+p_Z, and sweep R_X:R_Z for various code distances, defect rates, and bias values eta=p_Z/p_X. They report that a non-uniform schedule reduces the average logical error rate by up to 4.25x at 1% defect rate and 8.46x at 2% defect rate for d=13 at eta=5, and that the optimal ratio R* is essentially independent of code distance and defect rate, depending mainly on eta. They also extend the idea to defect-free codes under a crosstalk model and report up to 4.5x improvement.

Significance. If the claims hold, the paper identifies a practical, calibration-driven tuning knob for surface codes with fabrication defects, complementing code-level bias tailoring. The use of standard tools (Stim, PyMatching) and the SnL framework is appropriate, and the empirical finding that R* depends primarily on noise bias is interesting and potentially useful. The paper is clearly written and the crosstalk section is presented as an illustrative extension. However, the two central claims rest on a restrictive p_Y=0 noise model and on simulation studies with no statistical uncertainty quantification, both of which need to be addressed before the quantitative results can be relied upon.

major comments (3)
  1. [Section III (Pauli channel setup)] The simulation model sets p_Y=0 and asserts that Y errors 'flip both syndromes, so including them rescales the LER by a constant factor at fixed bias, without changing the trends observed.' This assertion is not derived and is inconsistent with Section IV-B, where the calibration relation eta = T1/T2 - 1/2 comes from a Pauli-twirled T1/T2 channel in which p_Y = p_X. Under a non-uniform schedule R_X:R_Z != 1, a Y error is a correlated XZ event whose X- and Z-syndrome components are sampled in alternating rounds at different rates; changing p_Y therefore alters the relative weight of the X and Z decoding subgraphs rather than only the overall error rate. The optimal ratio R* could shift with p_Y, so the headline improvements (4.25x at 1% defects, 8.46x at 2% defects) and the claim that R* depends only on eta are conditional on this unverified assumption. Please repeat the key sweeps (at least Figures 4 and 6) with nonzero p_Y, e.g., p_Y = p_X as implied by T1/T2 twirling, and report whether R* and the improvement factors change.
  2. [Sections IV-A and IV-B (statistical reporting)] No shot counts, error bars, or confidence intervals are reported for any LER estimate, and each configuration uses only 10 to 30 stochastic defect patches (Figure 4 uses 10 patches; Figure 5 uses 30 patches). The definition of R* in Section IV-A as the smallest ratio whose average LER is within 1% of the observed minimum, and the 5% bars in Figure 6, therefore cannot be interpreted without knowing the sampling noise. The claim that R* is essentially independent of code distance and defect rate is a null result that needs uncertainty quantification; with a coarse ratio grid and small patch counts, the observed flatness of R* versus d could be an artifact of sampling variation. Please report the number of shots per patch, the number of patches per point, and standard errors or confidence intervals for the LER and for R*.
  3. [Section IV-B (calibration-based selection)] The paper advertises that the optimal ratio can be selected directly from measured calibration data, without running device-specific simulations. However, the only quantitative output is Figure 6, which shows R* for eta in {1,2,4,6,8,10} and for discrete ratios up to 49:1, with no fitted curve, interpolation rule, or statement of how sensitive the LER is to a small error in R*. Since the transferability claim relies on the relationship R*(eta) being robust, the manuscript should provide either a table of recommended ratios, a fit R*(eta), or an explicit error tolerance (e.g., the range of R where LER is within 1% of the minimum) as a function of eta.
minor comments (6)
  1. [Section III (simulation setup)] The text 'performing O(d) rounds of syndrome extraction' is underspecified; state the exact total number of rounds used for each d and confirm that the total number of rounds is held fixed across all schedules so that LER comparisons are not confounded by different memory times.
  2. [Figures 4 and 8] The improvement at d=13, dr=1%, eta=5 is reported as 4.46x in Figure 4 and 4.25x in Figure 8, while the abstract gives 4.25x; please reconcile these numbers and state which patch set and ratio grid each value refers to.
  3. [Section IV-A (optimality tolerance)] The 1% tolerance used to define R* is not justified; please report how the choice of tolerance affects the identified R* (e.g., a sensitivity sweep) or provide shot-noise-based error bars that motivate the tolerance.
  4. [Section IV-B (calibration relation)] The relation eta = T1/T2 - 1/2 is stated without derivation; consider adding the Pauli-twirling expressions for p_X, p_Y, p_Z or a citation with the explicit formula to make the connection to Section III's p_Y=0 model transparent.
  5. [Section V (crosstalk model)] The crosstalk model sets the simultaneous-to-isolated gate error ratio alpha in [1,2] but cites only general randomized-benchmarking references; state which measured values or specific experimental results motivate this range.
  6. [General] There are minor typographical issues: 'X-ZRound' appears in the title line, 'eta=p z/px' should use proper subscripts, and the notation R_X:R_Z should be defined explicitly in the abstract or introduction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimal scheduling ratio is obtained by simulation sweeps, not by construction from the inputs, and the paper's claims are empirically self-contained.

full rationale

The paper's central claim — that under biased noise the optimal X-to-Z round scheduling ratio departs from 1:1 and is set primarily by the noise bias — is established by direct simulation: the LER is measured for a grid of ratios R_X:R_Z, the minimum is identified numerically, and the dependence on eta, distance, and defect rate is read off those sweeps. This is a posterior observation, not a quantity that is defined or fitted in terms of itself. The p_Y=0 noise-model choice is a stated input assumption, and while the assertion that Y errors merely rescale the LER is unproven, it is not a circular step: it does not define R*, and it is not used as a substitute for the simulation that produces the reported improvements. The only self-citation is reference [46] (Murali, McKay, Martonosi, Javadi-Abhari) used as background for the crosstalk model in Section V alongside external references [45] and [47]; that citation is not load-bearing for the main scheduling result, and the crosstalk model itself is explicitly adopted as a simple abstraction rather than derived from the cited work. The optimal-ratio-versus-bias relation is characterized empirically and could, in principle, have come out differently; therefore the derivation chain does not reduce to its inputs.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced. The scheduling ratio is a tunable parameter, not an invented entity. The ledger is dominated by simulation-modeling assumptions, with the p_Y=0 assumption the most fragile.

free parameters (1)
  • Optimality tolerance for defining R* = 1% of observed minimum average LER
    The paper defines the optimal schedule as the smallest R whose average LER lies within 1% of the observed minimum (Section IV-A). This hand-chosen threshold determines the reported R* values and contributes to the claimed independence of R* from code distance.
assumptions (6)
  • domain assumption Circuit-level Pauli noise with rate p on every gate, measurement, and idle location
    Standard simulation model; assumed representative of physical platforms.
  • ad hoc to paper Y errors can be set to zero because they flip both X and Z syndromes and therefore only rescale the LER by a constant factor at fixed bias
    Asserted in Section III without derivation or citation; central to isolating per-basis scheduling effects.
  • domain assumption Average LER over X- and Z-basis state preparations is the relevant figure of merit
    Used because a real computation has unknown logical basis; reasonable but not the only possible metric.
  • domain assumption Defects are independent per qubit and coupler at rate dr
    Standard stochastic defect model from prior work; simplifies sampling but may not capture spatial correlations in fabrication.
  • domain assumption SnL defect adaptation produces anti-commuting X/Z gauge checks requiring alternating rounds
    Imported from Leroux et al. [9]; the paper builds entirely on this construction.
  • ad hoc to paper Crosstalk can be modeled by an error-rate multiplier alpha applied to CNOTs executed in parallel, with alpha in [1,2]
    A synthetic model justified by randomized-benchmarking studies; not a validated physical model.

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Cite this review

Pith. "Pith review of X-Z Round Scheduling for the Surface Code with Defects under Biased Noise." pith.science (2026). https://pith.science/paper/QIDSYTVN

@misc{pith2026260805518,
  author       = {Pith},
  title        = {Pith review of: X-Z Round Scheduling for the Surface Code with Defects under Biased Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIDSYTVN}},
  note         = {Machine review of arXiv:2608.05518}
}
abstract

Fault-tolerant Quantum Computing (FTQC) relies on Quantum Error Correction (QEC) codes that encode logical qubits across many physical qubits to detect and correct errors. The surface code is among the most widely studied codes due to its high error threshold, the existence of efficient decoders, and hardware-friendly properties: a planar, two-dimensional layout with nearest-neighbor connectivity. In practice, however, the fabrication of solid-state quantum processors introduces hardware defects, resulting in defective qubits and couplers that must be discarded. Adapting the surface code to these defects often requires measuring the $X$- and $Z$-type checks in separate rounds rather than simultaneously. In this work, we investigate the optimal $X$-to-$Z$ checks round-scheduling ratio under biased noise systems. Our results characterize how key architectural parameters, such as noise bias, code distance, and defect rate, impact the logical error rate. We provide insights into how to determine the optimal scheduling ratio directly from device calibration data, enabling manufacturers to maximize performance without extensive simulations. Our approach reduces the logical error rate by up to $4.25\times$ at a $1\%$ defect rate and up to $8.46\times$ at a $2\%$ defect rate for a distance-$13$ surface code under moderately biased noise. Furthermore, we demonstrate that the benefits of round-scheduling extend beyond the defective-hardware setting. In biased-noise architectures subject to CNOT crosstalk, separating $X$ and $Z$ measurement rounds yields up to $4.5\times$ reduction in logical error rate.

Figures

Figures reproduced from arXiv: 2608.05518 by the authors.

Figure 1
Figure 1. (left) Distance-5 rotated surface code with d 2 data qubits and d 2 − 1 ancilla qubits which measure X- and Z-type stabilizers. (middle) Surface code under the presence of 1% defect rate, with a defective ancilla qubit, necessitating two super-stabilizers (one X, one Z) to obtain the syndrome. (right) Surface code under the presence of 2% defect rate, with one defective ancilla, one defective data qubit, and two def… view at source ↗
Figure 2
Figure 2. Probability of obtaining a non-defective processor as a function [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. (left) The X & Z decoding graph for the left patch depicted in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Logical error rate for defective systems ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Average logical error rate as a function of the scheduling ratio [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Optimal scheduling ratio R∗ as a function of code distance d across noise bias η ∈ [1, 10]. Vertical bars indicate the range of R values around R∗ over which the average LER stays within 5% of its observed minimum; these widen with η, showing reduced sensitivity to the…
Figure 7
Figure 7. Figure 7: Average LER as a function of the scheduling ratio [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: Average LER as a function of the scheduling ratio R for code distances from [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: (left) When X- and Z-type stabilizers are measured in the same round, each layer of the syndrome-extraction circuit executes one CNOT per stabilizer in parallel, leading to crosstalk errors between them. (right) Scheduling a single stabilizer type per round halves the …
Figure 10
Figure 10. Figure 10: (left) Average LER as a function of round scheduling. The central tick XZ denotes the baseline in which all stabilizers are measured in every round; points to its left correspond to 1:RX schedules and points to its right correspond to 1:RZ schedules. (right) LER impro…
Figure 11
Figure 11. Figure 11: Noise bias η on publicly available IBM Heron processors, extracted from reported T1 and T2 calibration data, via η ≈ T1/T2 − 1/2. The main panel shows the distribution of η over qubits for each device, and the inset reports the per-device mean. adapted to defects is a…

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.