Optimization of Quantum Error Correcting Code under Temporal Variation of Qubit Quality

Subrata Das; Swaroop Ghosh

REVIEW 3 major objections 4 minor 17 references

Optimization of Quantum Error Correcting Code under Temporal Variation of Qubit Quality

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Per-qubit, per-day code distance selection halves the physical-qubit cost of quantum error correction on 127-qubit devices.

desk verdict The adaptive-QEC idea is sensible, but the headline savings number is an artifact of an unfavorable baseline, so the paper needs a redesigned evaluation before its quantitative claims can be trusted. read the letter →

arxiv 2505.06165 v1 pith:DHL5AUU2 submitted 2025-05-09 quant-ph cs.ET

Subrata Das , Swaroop Ghosh This is my paper · ORCID

classification quant-phcs.ET PACS 03.67.Pp

keywords quantumerrorcorrectionsurfacecodeadaptivedistancetemporalvariationqubitqualityvariabilitycalibrationdatalogicalrateresourceoverhead

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

Qubit error rates on real superconducting processors drift from day to day and differ across qubits, so a single fixed surface-code distance is either too weak for noisy qubits or wasteful for stable ones. The paper claims that selecting the code distance per qubit and per calibration day, using the smallest distance whose simulated logical error rate still meets the target, avoids both failure modes. On 12 days of calibration data from a 127-qubit device, this adaptive assignment keeps 85-100% of qubits usable while cutting physical qubit overhead by over 50% per logical qubit; on two other 127-qubit devices the savings reach up to 71%. The practical point is that QEC can be configured from data already being collected, instead of reserving a worst-case error rate.

What carries the argument

The carrying object is the simulated logical-error-rate curve for rotated surface codes, which converts a physical Pauli-X error rate into a minimum code distance that still meets a target logical error rate such as $10^{-6}$. This curve supplies the per-distance cutoff thresholds used by the adaptive assignment rule: sort qubits by the day's calibration error, read off the smallest distance whose simulated logical error rate stays below the target, and drop qubits needing a distance above the practical maximum. The rule is what turns raw calibration data into a concrete resource allocation, and the quadratic qubit count of the rotated surface code, about $d^2$ qubits per logical qubit, is what makes the savings material.

What would settle it

Run the proposed assignment on a real device for one day: assemble distance-9 logical qubits from qubits whose daily Pauli-X error is near $10^{-3}$ and measure the logical error rate; the method predicts it should meet the $10^{-6}$ target, so observing a logical error rate substantially above that level would show the simulation thresholds are too optimistic for real hardware.

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Extended reading notes

Core claim

The central claim is that a resource-efficient QEC configuration should be a per-qubit, per-day decision rather than a global constant. Using daily Pauli-X calibration error rates, the paper simulates rotated surface codes of distances $d=3$ through $d=21$ to find, for each distance, the physical error threshold that still reaches a logical error rate of $10^{-6}$; the thresholds are $7\times10^{-4}$ for $d=7$, $10^{-3}$ for $d=9$, $2\times10^{-3}$ for $d=11$, and $7\times10^{-3}$ for $d=13$. The adaptive rule assigns each qubit the smallest distance meeting the target from that day's calibration, and excludes qubits whose error rate would require a distance above the allowed maximum (for example, above 9). Compared with a worst-case fixed distance-13 configuration using 169 physical qubits per logical qubit, the adaptive scheme with maximum distance 9 uses about 81 qubits per logical qubit while keeping roughly 85% of the device's qubits usable, giving the reported overhead reduction of over 50%, with up to 71% on two other devices where distance-7 can be used.

Load-bearing premise

The load-bearing assumption is that a qubit's daily Pauli-X calibration error rate can stand in for the full per-gate error probability used in the logical-error simulation, even though the simulation includes all gates, resets, and measurements while the calibration analysis uses only single-qubit Pauli-X values.

Editorial extensions

If this is right

QEC configuration can be refreshed from calibration data that hardware providers already publish, so day-to-day drift no longer requires a permanent worst-case code distance.
A compiler or scheduler can use the per-day distance map to place logical qubits only on qubits whose error rate supports the target, reducing wasted physical qubits.
On the devices studied, the adaptive rule keeps 80-100% of physical qubits available for encoding, so practical logical-qubit capacity does not collapse when a few qubits degrade.
High-error outlier qubits are excluded rather than encoded at prohibitive distances, concentrating the error budget on qubits that can actually meet the logical target.

Reading between the lines

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

The paper's thresholds come from a symmetric depolarizing model; extending the same calibration-driven assignment to CNOT error rates, noise bias, or correlated errors would likely shift the usable-qubit fractions and could make the savings device-specific.
A system-level scheduler might reasonably trade the logical error rate target for more logical qubits: relaxing the target from $10^{-6}$ to $10^{-5}$ would admit more qubits at distance 7 and could cut overhead further than reported.
The method's daily refresh assumes calibration remains valid for the whole execution window; a direct test would be to compare morning and evening calibrations on the same qubits and measure whether the assigned distance still meets the logical target.
The per-qubit independence assumption ignores crosstalk and competing resource demands; a global optimizer over all logical qubits simultaneously is a natural next step.

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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

Quick take: this is a sensible engineering idea with a misleading headline number. The >50% overhead saving is computed against a worst-case distance-13 baseline, so it mostly reflects the choice of maximum distance rather than the adaptive rule. A fixed distance-9 policy that excludes bad qubits would give the same usable-qubit fraction and the same 81 qubits per logical qubit. The paper never reports the actual distribution of assigned distances, so the advantage of day-by-day adaptation over a moderate fixed distance is unproven.

What is genuinely useful: the authors assemble 12 days of calibration data from ibm_kyiv, show that Pauli-X and CNOT error rates fluctuate significantly across qubits and time, and then run standard Stim/PyMatching simulations of rotated surface codes to extract per-distance thresholds. That part is reproducible and the direction of the argument is plausible. Applying the same procedure to two other 127-qubit devices is a nice consistency check. If a user is willing to reconfigure the QEC stack daily, per-qubit distance selection is a reasonable resource-management knob.

The soft spots are real and not minor. First, the baseline comparison is unfair. Baseline distance-13 is chosen as the worst-case over all qubits and days, while adaptive is capped at distance-9. That isolates the maximum-distance choice, not adaptivity. The 71% saving on Brisbane/Sherbrooke is just the ratio 169/49 for fixed distances; it has nothing to do with the adaptive rule. Second, the noise model mismatch: the calibration Pauli-X error rate is used directly as the depolarizing probability p in a circuit-level noise model that includes CNOTs, resets, and measurements. These are not the same quantity, and the paper does not justify the mapping. This could shift the thresholds and therefore the usable-qubit counts. Third, the evaluation choices (max distance 9, target 1e-6, cutoff 8e-3) look post hoc; there is no sensitivity analysis.

The paper should not be desk-rejected because the idea is testable and the data are real. But the current quantitative claims need major revision. The authors should report the actual histogram of distances assigned per day, compare adaptive assignment against a fixed distance-9 (and 11) policy with the same exclusion rule, and either validate the noise mapping or switch to a more honest per-gate error model.

I'd send this to peer review with a request for major revision. It is a solid workshop-level contribution as is, but with a redesigned evaluation it could be a genuinely useful reference for NISQ-era resource optimization.

Referee Report

3 major / 4 minor

Summary. The paper analyzes 12 days of calibration data from IBM's 127-qubit device ibm_kyiv, and additional data from ibm_brisbane and ibm_sherbrooke, showing temporal and spatial variation in Pauli-X and CNOT error rates. It proposes an adaptive QEC strategy that selects a rotated surface-code distance per qubit and per day, using Stim/PyMatching simulations to determine the smallest distance that meets a target logical error rate of 10^-6 and excluding qubits that would require a distance above an allowed maximum. The central claim is that this adaptive assignment reduces physical-qubit overhead by more than 50% per logical qubit on ibm_kyiv and up to 71% on the other two devices, while preserving access to 80-100% of usable qubits.

Significance. If the quantitative claims were properly supported, the paper would make a useful practical contribution: dynamic, calibration-aware selection of code distance is a plausible resource-optimization strategy for near-term QEC, and the paper grounds it in real device data rather than synthetic noise models. The strengths are the use of publicly available calibration data, the concrete pipeline of distance assignment from simulated logical-error thresholds, and the explicit target logical error rate. However, the headline savings are currently computed against a comparison that isolates the maximum allowed distance rather than adaptivity, and the noise-model mapping from calibration Pauli-X rates to the circuit-level depolarizing simulation is not justified. These issues are load-bearing for the central claim, so the paper needs substantial revision before the results can be accepted.

major comments (3)

[Section IV.B] The reported 52% saving on ibm_kyiv compares a worst-case distance-13 baseline (169 physical qubits per logical qubit) with an adaptive scheme capped at distance-9 (81 physical qubits). This comparison measures the choice of maximum distance, not the benefit of day-by-day, per-qubit adaptation: a fixed distance-9 policy that simply excludes qubits with error rate above the distance-9 threshold would yield the same usable-qubit fraction (~85%) and the same 81 physical qubits per logical qubit. The same issue affects the ibm_brisbane/ibm_sherbrooke result, where the 71% saving is essentially the ratio 169/49 from choosing distance-7 instead of distance-13. To support the claim that adaptation creates the savings, the paper should report the actual distribution of assigned distances per day, the average physical-qubit overhead per logical qubit under the adaptive rule, and a comparison against fixed-distance policies at each candidate distance (e.g., d=7, d=9, d=11) with the same qubit-exclusion rule.
[Section III.A and III.B] The paper states that 'we restrict our analysis and logical error simulations to only single-qubit Pauli-X error rates' (Section III.A), but the simulation in Section III.B is described as a circuit-level model with symmetric depolarizing errors applied before and after Clifford gates, resets, and measurements. Since the calibration data shows CNOT error rates in the 10^-2 to 10^-1 range, substantially above the simulated thresholds quoted in Section IV.B, the mapping from the daily Pauli-X calibration value to the per-gate depolarizing probability p in Stim is not justified. If the simulation includes CNOT errors at the same p, then the thresholds cannot be correct for the real hardware; if it does not, the claim that the simulated thresholds govern usability of real qubits is unsupported. The authors should either simulate a noise model that actually uses only the single-qubit Pauli-X error rates with no CNOT/readout errors, or justify explicitly why excluding CNOT, resets, and measurement errors does not change which qubits are classified as usable.
[Section IV.B and Fig. 5] The threshold values used for qubit usability (7x10^-4 for d=7, 10^-3 for d=9, 2x10^-3 for d=11, 7x10^-3 for d=13) are read from the simulated curves in Fig. 4, but the paper does not report the number of simulation shots, the statistical uncertainty in the logical error rates, or the exact criterion used to extract each threshold from the crossing of the p_L = 10^-6 line. Because many calibration error-rate values in Fig. 3(a) lie very close to these thresholds, small simulation uncertainties could change the day-by-day usable-qubit percentages and hence the final overhead savings. The authors should provide error bars or confidence intervals on the simulated logical error rates and state how the thresholds were obtained.

minor comments (4)

[Section I] There is a typo in the first paragraph of the introduction: 'In the the rest of the paper' should be 'In the rest of the paper'.
[Section II.A] There are formatting issues in the text: 'distanced typically requires' should be 'distance d typically requires', and mathematical expressions such as 'd2 qubits' and '132 = 169' should be typeset properly as d^2 and 13^2 = 169 respectively.
[Section IV.B] The text '92 = 81 physical qubits' should be written as 9^2 = 81 to avoid confusion.
[Section III.B] The description of the Stim simulation should state which gates are included (e.g., Hadamard, S, CNOT, measurements, resets) and whether the depolarizing error rate for single-qubit gates equals that for two-qubit gates; this information is needed to reproduce the thresholds.

Circularity Check

1 steps flagged · score 2.0 of 10

Headline overhead saving is a ratio of two chosen distances, not a measured benefit of adaptive QEC; thresholds themselves are independent.

other [Section IV.B (Results and Trade-offs), baseline/adaptive overhead comparison]
"Baseline QEC: A single worst-case error rate across all qubits and days would force us to design for the maximum code distance needed, in this case, distance 13. This requires 13^2 = 169 physical qubits per logical qubit. • Adaptive QEC: We exclude high-error qubits and use smaller distances when possible. For instance, with a maximum distance of 9, we can still use approximately 85% of the qubits (about 108 out of 127). Each logical qubit then uses only 9^2 = 81 physical qubits. This leads to a resource saving of roughly 52% per logical qubit."

The advertised 52% saving is just 1 - (9/13)^2 = 0.52. Both numbers are chosen inputs of the comparison, not outputs of the adaptive algorithm or of the Stim simulation. With a maximum distance of 9, any policy that excludes qubits above the d=9 threshold—including a fixed distance-9 code—uses at most 81 physical qubits per logical qubit and retains the same ~85% usable-qubit fraction. The same structure applies to the 71% claim on the other two devices, which is 1 - (7/13)^2. The headline overhead reduction therefore reduces by construction to the ratio of the selected baseline distance and selected cap distance, and does not measure the temporal/per-qubit adaptivity that the paper claims to validate.

full rationale

The logical-error-rate thresholds (7e-4, 1e-3, 2e-3, 7e-3) are produced by a Stim/PyMatching circuit-level simulation with a stated depolarizing model; they are not fit to ibm_kyiv calibration data, and no parameter of the simulation is tuned to reproduce the paper's overhead claims. The proposed selection rule then simply compares each qubit's measured Pauli-X error rate to these externally generated thresholds, so the usability classification and distance assignment are not circular. The self-citations [7] and [15] are illustrative (ancilla-count figure, Stim usage) and not load-bearing; the claim of efficient QEC rests on the independent simulation. The only step that reduces by construction is the headline 52%/71% overhead saving: it is the ratio of chosen squared distances (81/169 and 49/169) rather than a measured property of the adaptive rule. A fixed-distance d=9 or d=7 policy with the same qubit exclusion would give the same numbers. This is a baseline-comparison artifact, not a fitting loop, so I rate it as a minor non-load-bearing circularity (score 2).

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The simulation thresholds are not fitted to the calibration data, so the main claim is not circular in the narrow sense. The reported savings, however, depend on evaluation choices made with knowledge of the data, and the usability classification assumes a direct mapping from Pauli-X calibration error rates to circuit-level depolarizing noise. No new physical entities are introduced.

free parameters (4)

Target logical error rate p_L,target = 1e-6
Chosen as the design target from cited large-scale QEC proposals; it sets all usability thresholds and therefore directly determines the reported overhead savings.
Maximum allowed code distance d_max = 9
Introduced in Section IV.A as the cutoff above which qubits are excluded; no sensitivity analysis is given and this choice directly determines the 52% savings figure.
Baseline code distance = 13
Selected in Section IV.B as the worst-case fixed distance; the selection rule is not defined precisely and this choice defines the denominator in the overhead saving calculation.
Physical error cutoff for QEC = about 8e-3
Read from Fig. 4 as the point where the 1e-6 target becomes unreachable at reasonable distances; used to declare qubits unusable above this error rate.

assumptions (6)

domain assumption Surface-code logical error scaling follows Eq. (1): p_L is roughly alpha times (p/p_th) raised to (d+1)/2.
Standard surface-code result cited to [3], [6]; used throughout to motivate distance selection and threshold logic.
domain assumption A circuit-level symmetric depolarizing noise model in Stim, with errors before and after Clifford gates, resets, and measurements, captures the relevant hardware failure modes for rotated surface codes.
Assumed in Section III.B; not validated against the IBM calibration data or any hardware experiment.
ad hoc to paper A qubit's daily Pauli-X calibration error rate equals the per-gate depolarizing error probability p in the Stim simulation.
This mapping is never justified; the calibration metric measures single-qubit gate error under different conditions than the circuit-level depolarizing model.
ad hoc to paper Excluding CNOT, measurement, and readout errors does not change which qubits are usable at a given code distance.
Acknowledged in Section III.A as a restriction; these errors are major contributors in surface code syndrome extraction and could change the usable-qubit classification.
domain assumption The target logical error rate of 1e-6 is an appropriate design threshold for practical fault-tolerant computation.
Cited to large-scale proposals; reasonable but not derived in this paper.
standard math A rotated surface code of distance d requires about d squared physical qubits per logical qubit.
Standard layout fact cited to [15]; used to compute overhead savings.

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Cite this review

Pith. "Pith review of Optimization of Quantum Error Correcting Code under Temporal Variation of Qubit Quality." pith.science (2026). https://pith.science/paper/DHL5AUU2

@misc{pith2026250506165,
  author       = {Pith},
  title        = {Pith review of: Optimization of Quantum Error Correcting Code under Temporal Variation of Qubit Quality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DHL5AUU2}},
  note         = {Machine review of arXiv:2505.06165}
}

read the original abstract

Error rates in current noisy quantum hardware are not static; they vary over time and across qubits. This temporal and spatial variation challenges the effectiveness of fixed-distance quantum error correction (QEC) codes. In this paper, we analyze 12 days of calibration data from IBM's 127-qubit device (ibm_kyiv), showing the fluctuation of Pauli-X and CNOT gate error rates. We demonstrate that fixed-distance QEC can either underperform or lead to excessive overhead, depending on the selected qubit and the error rate of the day. We then propose a simple adaptive QEC approach that selects an appropriate code distance per qubit, based on daily error rates. Using logical error rate modeling, we identify qubits that cannot be used and qubits that can be recovered with minimal resources. Our method avoids unnecessary resource overhead by excluding outlier qubits and tailoring code distances. Across 12 calibration days on ibm_kyiv, our adaptive strategy reduces physical qubit overhead by over 50% per logical qubit while maintaining access to 85-100% of usable qubits. To further validate the method, we repeat the experiment on two additional 127-qubit devices, ibm_brisbane and ibm_sherbrooke, where the overhead savings reach up to 71% while still preserving over 80% qubit usability. This approach offers a practical and efficient path forward for Noisy Intermediate-Scale Quantum (NISQ)-era QEC strategies.

Figures

Figures reproduced from arXiv: 2505.06165 by the authors.

**Figure 2.** Representation of distance-3 surface codes. Unrotated surface code [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** (a) Pauli-X error rate across 12 calibration days for all 127 qubits in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Percentage of usable qubits on each calibration day across four code [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 4.** Figure 4: Logical error rate as a function of physical Pauli-X error rate for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 6.** Figure 6: Percentage of usable qubits across seven calibration days for two addi [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

An introduction to quantum error correction,

D. Gottesman, “An introduction to quantum error correction,” in Pro- ceedings of Symposia in Applied Mathematics , vol. 58, 2002, pp. 221– 236

2002

[2] [2]

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2018

[3] [3]

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A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Physical Review A—Atomic, Molecular , and Optical Physics , vol. 86, no. 3, p. 032324, 2012

2012

[4] [4]

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D. S. Wang, A. G. Fowler, and L. C. Hollenberg, “Surface code quantum computing with error rates over 1%,” Physical Review A—Atomic, Molecular , and Optical Physics , vol. 83, no. 2, p. 020302, 2011

work page 2011

[5] [5]

Topological quantum memory,

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,” Journal of Mathematical Physics , vol. 43, no. 9, pp. 4452– 4505, 2002

2002

[6] [6]

Quantum error correction for quantum memories,

B. M. Terhal, “Quantum error correction for quantum memories,” Reviews of Modern Physics , vol. 87, no. 2, pp. 307–346, 2015

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[7] [7]

Quantum prometheus: Defying overhead with recycled ancillas in quantum error correction,

A. Chatterjee, A. Ghosh, and S. Ghosh, “Quantum prometheus: Defying overhead with recycled ancillas in quantum error correction,” arXiv preprint arXiv:2411.12813, 2024

arXiv 2024

[8] [8]

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A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, and J. M. Gambetta, “Validating quantum computers using randomized model circuits,” Physical Review A , vol. 100, no. 3, p. 032328, 2019

2019