Exponential logical-error reduction in quantum memories via optimal syndrome-measurement timing

Kishor Bharti; Leandro Aolita; Tobias Haug

REVIEW 4 major objections 5 minor 56 references

Exponential logical-error reduction in quantum memories via optimal syndrome-measurement timing

T0 review · 4 major / 5 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 For quantum memories, the optimal syndrome-measurement interval scales inversely with code distance, and this choice reduces logical error rates exponentially compared with any distance-independent schedule.

desk verdict A practical, honest phenomenological analysis of syndrome-timing optimization whose main claims are conditional on a fitted ansatz but deserve referee attention. read the letter →

arxiv 2608.06242 v1 pith:HGS2ESA4 submitted 2026-08-06 quant-ph

Tobias Haug , Kishor Bharti , Leandro Aolita This is my paper · ORCID

classification quant-ph PACS 03.67.Pp

keywords quantumerrorcorrectionsurfacecodesyndromemeasurementtiminglogicalratememoryadaptivephenomenologicalnoisemodeldistancescaling

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 treats the time between syndrome measurements of a quantum memory as a tunable control parameter rather than a fixed clock cycle. It proposes a phenomenological formula for the logical error rate per unit time and shows that the optimal interval shrinks as $1/d$ with the code distance $d$, yielding a logical error rate that is exponentially lower in $d$ than any distance-independent interval. For time-dependent noise, it develops an adaptive scheme that shortens the interval when syndrome activity rises, and shows this outperforms every fixed-interval protocol for short, strong noise bursts. If correct, existing surface-code memories can lower their logical error rates at no additional qubit or gate cost, simply by retiming syndrome measurements.

What carries the argument

The load-bearing object is the phenomenological logical-noise ansatz of Eq. (5), which encodes the trade-off: the prefactor $1/\Delta t$ and base noise $p$ capture faults from the measurement circuits themselves, while the factor $(1+\lambda\Delta t)^{g(d+1)/2}$ captures idle-time errors accumulated between measurements. The exponent fraction $g\in[0,1]$ (fitted as $g\approx 0.8$ for read-out noise coefficient $b_{\rm read}=1$) determines how much of the logical-error exponent is sensitive to waiting time, and it is the source of both the $1/d$ optimum and the exponential advantage. For the adaptive extension, the central mechanism is a per-round log-likelihood ratio of measured syndrome activity, averaged over a window, that triggers shorter intervals when a noise burst is detected.

What would settle it

Run a surface code memory at two or more distances under the same physical noise with known $\lambda$, sweep $\Delta t$, and locate the minimum of logical error rate per unit time; if the optimal interval does not move as $1/d$, or the ratio of errors at fixed $\Delta t$ to optimal $\Delta t$ does not grow exponentially in $d$, the central claim is refuted.

Watch

Extended reading notes

Core claim

The central claim is that the logical error rate per unit time of a rotated surface-code memory under phenomenological noise is captured by the ansatz $R = (1/\Delta t)(A/d^\beta)(p/p_{\rm th})^{(d+1)/2}(1+\lambda\Delta t)^{g(d+1)/2}$. Minimizing this expression over the syndrome interval gives $\Delta t^\star \approx 2/(\lambda g d)$, so the optimal interval is inversely proportional to code distance. Substituting this optimum yields an improvement over any fixed interval of $\Gamma_{\rm opt} \sim (1/e)(\Delta t^\star/\Delta t)\exp(\Delta t/\Delta t^\star)$, an exponential reduction in $d$ for large distances. The paper also shows that for time-dependent idle noise, an adaptive controller based on a log-likelihood ratio of syndrome activity achieves a logical error rate lower than any fixed-interval schedule, with advantage scaling as $r/(e\log r)$ for bursts of strength $r$.

Load-bearing premise

The whole analysis assumes the logical error rate follows the power-law form $R=(1/\Delta t)(A/d^\beta)(p/p_{\rm th})^{(d+1)/2}(1+\lambda\Delta t)^{g(d+1)/2}$; the exponent factor is fit from Monte Carlo data rather than derived, and if real devices have a different dependence on waiting time the $1/d$ optimum and exponential advantage would not hold.

Editorial extensions

If this is right

If the ansatz holds, any surface-code memory can reduce its logical error rate per unit time by setting $\Delta t = 2/(\lambda g d)$ instead of a fixed interval; the reduction grows exponentially with $d$.
For the noise parameters of a recent superconducting surface-code experiment, the model predicts up to $40\%$ reduction in logical error rate per unit time at distances $3,5,7$ by using longer intervals than the implemented one.
For larger distances than those demonstrated, the optimized interval shortens, and the effective noise-suppression factor increases by about $24\%$.
For time-dependent noise, the adaptive syndrome-activity controller outperforms every fixed-interval protocol, with gains up to a factor of about $1.95$ for distance $15$ under burst noise.

Reading between the lines

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

A testable extension is to wire the adaptive controller into decoders that already use soft information from read-out; earlier burst detection could close the gap between the ideal-controller bound and the finite-window results.
Because the optimal interval is set by $\lambda$, devices with longer coherence times can wait longer between measurements, so the relative benefit of the $1/d$ rule is platform-specific; the same functional form should transfer to ion-trap and neutral-atom memories where idle errors are dominated by dephasing.
The exponential advantage assumes the interval can be shortened without raising measurement error; if faster measurements degrade read-out fidelity, a crossover distance exists beyond which the optimal timing is pinned at the hardware floor, and the exponential gain saturates.
The paper's asymmetry result implies that when hardware forces a choice, erring on the side of measuring too often is much safer than measuring too rarely, because the penalty is linear instead of exponential.

Share X Bluesky LinkedIn Reddit HN

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

Short version: the paper identifies a real operational knob for surface-code memories — how long to wait between syndrome rounds — and gives a clean phenomenological model for optimizing it. The main claims are conditional on a fitted ansatz, but the paper is upfront about that, and the numerical support is broad enough that the claims deserve to be evaluated by a qualified referee.

What is genuinely new: prior work treated the number of measurement rounds; this paper treats the physical interval as a continuous parameter and derives a distance-dependent optimum Δt* ∝ 1/d. Given the ansatz, the exponential reduction in logical error rate compared to distance-independent intervals follows from elementary minimization. The adaptive protocol for noise bursts, based on a log-likelihood ratio of syndrome activity, is also new and comes with a formal bound on false-positive/false-negative rates that scales as 1/(W d^2). The Monte Carlo tests cover a decent range of d, p, λ, and b_read, and the ansatz matches well with only a few fitted parameters. The Google Willow mapping is rough but useful, and the predicted ~40% reduction for d=3,5,7 is an actionable statement.

The soft spots are proportional. All the analytical results are consequences of the power-law ansatz in Eq. (5); the exponent g≈0.8 is fitted, not derived, and the paper provides no confidence interval or alternative-model comparison. The exponential advantage in d would vanish if the true idling dependence near the optimum were, say, exponential in λΔt with a different d-scaling. The authors are aware of this: they explicitly call the model phenomenological. The simulations also lack error bars and quantitative fit statistics, and in some supplemental figures A is refit per (d,p), which weakens the claim that the ansatz is predictive. The practical adaptive gains in Fig. 3 — a factor 1.26 and 1.95 — are more modest than the ideal closed-form bound, and the detection protocol's performance depends on optimized thresholds, so it is not yet a turnkey recipe.

None of these are fatal. The paper makes a clear, falsifiable prediction about timing choices that experiments can test. The main fix a referee should demand is uncertainty quantification for the fitted parameters and a sharper statement about where the ansatz might break down. I would bring this to a reading group and cite it for the timing-optimization idea. Send it to peer review; a serious referee can handle the ansatz question.

Referee Report

4 major / 5 minor

Summary. The paper treats the syndrome-measurement interval of a quantum memory as an optimizable control parameter and proposes a phenomenological logical-noise model, Eq. (5), of the form R = (1/Δt)(A/d^β)(p/p_th)^((d+1)/2)(1+λΔt)^{g(d+1)/2}. Minimizing this expression over Δt yields an optimal interval Δt* ≈ 2/(λ g d) that scales inversely with the code distance, and the paper argues that this choice gives an exponential-in-d reduction of the logical error rate relative to any distance-independent interval. The model is verified against Monte Carlo simulations of rotated surface codes under a phenomenological bit-flip noise model, including different distances, physical error rates, idling rates, and read-out noise coefficients. The paper also develops an adaptive timing scheme for time-dependent noise bursts, based on a syndrome-activity log-likelihood ratio, and maps the Google Willow calibration data to predict up to 40% reductions in logical error rate per unit time.

Significance. If the phenomenological ansatz faithfully describes the trade-off between idling errors and measurement-induced faults, the paper delivers a simple, hardware-agnostic optimization that applies to every surface-code memory: choose Δt ∝ 1/d to obtain an exponential improvement over fixed-interval schedules. This is a practically relevant result, and the adaptive extension addresses a genuine problem of non-stationary noise. The analytic minimization is correct given the ansatz, and the simulations cover a wide parameter range. The main weaknesses are that the ansatz is postulated rather than derived, the key exponent g is fitted to the very same simulations used for validation, and the numerical support lacks error bars and quantitative fit statistics. The qualitative 1/d scaling and exponential advantage do not depend on the fitted value g≈0.8 (they already hold for the standard g=1 formula), but the quantitative predictions, including the Google-Willow projection, do inherit the fitting uncertainty.

major comments (4)

[Eq. (5) and SM B] The central analytic results, Δt* ≈ 2/(λ g d) in Eq. (6) and the exponential advantage in Eq. (7), are algebraic consequences of the postulated ansatz (5), in particular of the factor (1+λΔt)^{g(d+1)/2}. The numerical validation in SM B fits A (and in Fig. 7 also g) to the same Monte Carlo data used for validation, and no error bars, sample counts, or goodness-of-fit statistics are reported. Please provide Monte Carlo confidence intervals for the fitted parameters, especially g, and a quantitative model comparison against the g=1 standard formula (for example, a likelihood-ratio test or ΔBIC) to justify the extra parameter.
[SM G and Fig. 2a] The main claim that Δt* ∝ 1/d is verified by a single linear fit of 1/Δt* versus d (η=0.402 in SM G) with no reported uncertainty, residuals, or goodness-of-fit. Because this scaling is the central result, please report the fit confidence interval and, if possible, compare the slope with the model prediction 2/(λ g) using the independently calibrated λ and g.
[Fig. 3 and SM F] The adaptive-detection protocol is demonstrated on only two parameter sets (d=7 and d=15), and the reported advantages Γadapt ≈ {1.26, 1.95} have no error bars or statistical significance assessment. In addition, the ideal bound in Eq. (10) assumes instantaneous knowledge of λ(t), while the practical controller suffers detection latency; please quantify the gap between the ideal and implemented protocols and describe how the hyperparameters (θ, W, hold-time) were optimized.
[SM H and Table II] The Google-Willow mapping uses a sensitivity-weighted error budget to assign p/p_th ≈ 0.293 and λ ≈ 0.31, and then computes optimal intervals and improvements assuming g=0.8. These quantitative predictions (up to 40% reduction and the 24% suppression-factor increase in SM I) depend on the mapping choices and on the fitted g. Please add a sensitivity analysis that shows how Γopt varies when g is varied within its uncertainty and when the error-budget assignment is changed, so that the experimental prediction is not presented as a point estimate.

minor comments (5)

[SM B] The main text states that g≈0.8 is stable for b_read=1, but SM B reports g=0.85 for b_read=0.5 and g=0.8 for b_read=2; please clarify whether g is a fitted parameter for each read-out setting and list the fitted values with uncertainties.
[Eq. (11)] The log-likelihood ratio in Eq. (11) uses P(D_i | λ, Δt) and N_c without defining D_i; the definition appears later in SM F as the detector bit D_{j,c}. Please define D_i in the main text for self-containedness.
[Eq. (12)] The scaling P_FP, P_FN = O(1/(W d^2)) is stated as a headline result, but the explicit bound in SM F involves the detection separation Δμ and the constant κ; please state the dependence on these quantities in the main text or in the statement of Eq. (12).
[Introduction, first paragraph] There is a grammatical error: 'making decoder more likely to fail' should read 'making the decoder more likely to fail.'
[Fig. 2b] The fitted exponents γ={0.11,0.25} in Γopt ∝ exp(γd) are not connected to the model prediction from Eq. (7), which would give γ ≈ (g/2) log(1+λΔt) for the two displayed Δt values; please clarify the relation or the reason for the discrepancy.

Circularity Check

1 steps flagged · score 6.0 of 10

The central 1/d optimum and exponential advantage are algebraic consequences of the fitted ansatz (5), with exponent g≈0.8 fit to the same Monte-Carlo curves used as validation.

fitted input called prediction [Main text, 'Phenomenological noise model' and 'Exponential advantage from optimal intervals', Eqs. (5)–(7); SM B, SM G.]
"We propose the following ansatz R= 1/∆t A/d^β (p/p_th)^((d+1)/2) (1+λ∆t)^(g(d+1)/2) (5) ... In fact, for b_read=1, we usually only need to fit A, while the fit has stable β≈2 and g≈0.8. ... With our model, we now compute the optimal syndrome measurement interval ... ∆t⋆ = argmin_∆t R(∆t) = 2/(λ(g(d+1)−2)) ∼ 2/(λgd) (6) ... Thus, by scaling the syndrome interval inversely with distance, i.e. ∆t⋆ ∝1/d, we gain an exponential reduction in logical error rate compared to any distance-independent interval ∆t, which is the main result of our work."

The central result is obtained by differentiating the fitted ansatz: from Eq. (5), d/d∆t log R = αλ/(1+λ∆t) − 1/∆t with α=g(d+1)/2, so Eq. (6) is the stationary point of the fitted curve, and Eq. (7) is the ratio of the same fitted curve at two points. Neither is an independent consequence of the decoder model; the power-law factor (1+λ∆t)^{g(d+1)/2} is the input, with g≈0.8 fit to the Monte-Carlo data in SM B (and A refit per d,p). Calling the resulting ∆t*∝1/d and exponential Γ_opt 'predictions' is therefore a fitted-input-called-prediction: the optimum is statistically forced by the functional form chosen for the fit. The in-sample agreement in SM Figs.

full rationale

The prediction of ∆t*∝1/d and the exponential reduction Γ_opt are obtained purely by minimizing the fitted logical-error ansatz (5); no decoder-level derivation independent of that fit is given. The exponent g=0.8, which sets the location of the optimum and the advantage exponent, is itself fitted to the Monte-Carlo curves used for validation, without confidence intervals or model comparison against g=1. This is a fitted-input-called-prediction. However, the paper does provide full Monte-Carlo surface-code simulations (SM A) that match the ansatz over a wide parameter range, and Fig. 2b independently shows exponential improvement for the simulated data, so the central claim has genuine numerical support beyond the fit. The circularity is partial: the analytic derivation is not self-contained, but the numerical verification is not merely a refit of the claimed advantage. Score 6 reflects that the flagship analytic result reduces to a property of the fitted ansatz, while acknowledging the independent Monte-Carlo confirmation that prevents a higher score.

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

The central results are derived from a fitted phenomenological ansatz. The trade-off exponent g is the most sensitive input: g=0 would eliminate the exponential advantage, so the main prediction inherits its form from g. The noise model and threshold scaling are standard; the Google-Willow parameters are mapped from the experiment's error budget.

free parameters (4)

A = varies by p,d (e.g. 0.75 to 1.9 in SM B)
Nonuniversal fitting prefactor in ansatz (5), fit to Monte Carlo simulation of the rotated surface code; it multiplies the logical-error rate and does not affect the optimal interval.
g = 0.8 for b_read=1, 0.85 for b_read=0.5
Fraction of the leading logical exponent sensitive to idling noise in Eq. (5). This parameter controls the strength of the trade-off and the exponential advantage; it is fit from the same simulation data used for validation.
beta = 2
Scaling factor in ansatz (5), fixed from fits to simulation data; independent of Delta t, so it does not change the optimal interval.
p_th = 0.029 (b_read=1), 0.036 (b_read=0.5), 0.023 (b_read=2)
Code threshold fitted by crossing-point analysis for each read-out noise coefficient in SM C; for b_read=1 it is taken from Ref. [38].

assumptions (4)

standard math Standard surface-code logical error scaling R proportional to (p/p_th)^((d+1)/2) for constant measurement interval
Invoked as the g=1 limit of ansatz (5); sourced from Refs. [29,33-35].
domain assumption Phenomenological noise model: idling errors with probability 1-exp(-p lambda Delta t), measurement faults with p_stab = p, and read-out flips with b_read p
Assumed in Eqs. (1)-(3); it abstracts the physical error processes and follows the standard phenomenological model of Ref. [29].
ad hoc to paper The logical-error ansatz R in Eq. (5) holds over the simulated range of d, p, lambda, Delta t with fixed beta and g
This is the load-bearing model assumption. The exponential improvement and the 1/d scaling of the optimum are mathematical consequences of this power-law form. It is numerically supported but not derived from first principles.
domain assumption For adaptive protocols, the controller knows lambda(t) or can detect bursts from syndrome activity via the likelihood statistic (11)
The analytic adaptive advantage (10) assumes an ideal controller with instantaneous knowledge of lambda(t); the practical detector introduces latency and detection errors.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Exponential logical-error reduction in quantum memories via optimal syndrome-measurement timing." pith.science (2026). https://pith.science/paper/HGS2ESA4

@misc{pith2026260806242,
  author       = {Pith},
  title        = {Pith review of: Exponential logical-error reduction in quantum memories via optimal syndrome-measurement timing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HGS2ESA4}},
  note         = {Machine review of arXiv:2608.06242}
}

abstract

Syndrome-measurements timing is usually treated as a fixed clock cycle of a quantum error-correcting code. For quantum memories, however, the intra-measurement interval is itself an optimizable control parameter: measuring too rarely allows idling errors to accumulate, whereas measuring too often introduces measurement-induced faults. We propose a phenomenological logical-noise model for this trade-off and analytically show that the optimal syndrome-measurements interval scales inversely proportionally with the code distance and that this produces an exponential reduction of logical-error rates in the distance relative to constant-interval schedules. Furthermore, for time-dependent idling noise, we develop an adaptive timing strategy based on the measured syndrome activity that outperforms every fixed-interval protocol, with largest gains for short but strong noise bursts. Simulations of rotated surface-code memories with matching decoding validate the phenomenological model, the distance-dependent optimum, and the adaptive-strategy improvement. Moreover, with the experimental noise parameters reported by Google in Nature 638 (2025), our model predicts reductions in logical-error rates per unit time of up to $40\%$.

Figures

Figures reproduced from arXiv: 2608.06242 by the authors.

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4. Rotated surface code for [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: FIG. 5. Total logical error rate [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Total logical error rate [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7. Total logical error rate [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. Fit of code threshold for different read-out noise coefficient [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: FIG. 9. Optimal syndrome interval [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 37 canonical work pages

[1]

D. A. Lidar and T. A. Brun,Quantum error correction (Cambridge university press, 2013)

2013
[2]

B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics87, 307 (2015)

2015
[3]

D. J. Spencer, S. P. Jain, A. Tanggara, Z. Sun, T. Haug, D. Khu, and K. Bharti, Quantum error cor- rection and fault tolerance: A comprehensive tutorial, arXiv:2605.29137 10.48550/arXiv.2605.29137 (2026)

work page Pith review arXiv doi:10.48550/arxiv.2605.29137 2026
[4]

C. A. Pattison, M. E. Beverland, M. P. da Silva, and N. Delfosse, Improved quantum error correction us- ing soft information, arXiv preprint arXiv:2107.13589 10.48550/arXiv.2107.13589 (2021)

arXiv doi:10.48550/arxiv.2107.13589 2021
[5]

Márton and J

Á. Márton and J. K. Asbóth, Optimal number of stabi- lizer measurement rounds in an idling surface code patch, Quantum9, 1767 (2025)

work page 2025
[6]

Maurya and S

S. Maurya and S. Tannu, Synchronization for fault- tolerant quantum computers, inProceedings of the 52nd Annual International Symposium on Computer Architec- ture(2025) pp. 1370–1385

work page 2025
[7]

P. S. Mundada, A. Khindanov, Y. Wang, C. L. Ed- munds, P. Coote, M. J. Biercuk, Y. Baum, and M. Hush, Heterogeneous architectures enable a 138x reduction in physical qubit requirements for fault-tolerant quantum computing under detailed accounting, arXiv preprint arXiv:2604.06319 10.48550/arXiv.2604.06319 (2026)

arXiv doi:10.48550/arxiv.2604.06319 2026
[8]

S. M. Girvin, Introduction to quantum error correction and fault tolerance, SciPost Physics Lecture Notes , 070 (2023)

work page 2023

Show all 56 references

[9]

C. D. Wilen, S. Abdullah, N. Kurinsky, C. Stanford, L. Cardani, G. d’Imperio, C. Tomei, L. Faoro, L. Ioffe, C. Liu,et al., Correlated charge noise and relaxation er- rors in superconducting qubits, Nature594, 369 (2021)

2021
[10]

McEwen, L

M. McEwen, L. Faoro, K. Arya, A. Dunsworth, T. Huang, S. Kim, B. Burkett, A. Fowler, F. Arute, J. C. Bardin,et al., Resolving catastrophic error bursts from cosmic rays in large arrays of superconducting qubits, Nature Physics18, 107 (2022)

2022
[11]

McEwen, K

M. McEwen, K. C. Miao, J. Atalaya, A. Bilmes, A. Crook, J. Bovaird, J. M. Kreikebaum, N. Zobrist, E. Jeffrey, B. Ying,et al., Resisting high-energy impact events through gap engineering in superconducting qubit arrays, Physical Review Letters133, 240601 (2024)

2024
[12]

Bratrud, S

G. Bratrud, S. Lewis, K. Anyang, A. Colón Cesaní, T. Dyson, H. Magoon, D. Sabhari, G. Spahn, G. Wagner, R. Gualtieri,et al., Measurement of correlated charge noise in superconducting qubits at an underground facil- 6 ity, Nature Communications16, 9906 (2025)

2025
[13]

Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature638, 920 (2025)

2025
[14]

V. D. Kurilovich, G. Roberts, L. S. Martin, M. McEwen, A. Eickbusch, L. Faoro, L. B. Ioffe, J. Atalaya, A. Bilmes, J. M. Kreikebaum,et al., Correlated phase error bursts in a gap-engineered superconducting qubit array, Physical Review X16, 021025 (2026)

2026
[15]

Tanggara, M

A. Tanggara, M. Gu, and K. Bharti, Strategic code: A unified spatio-temporal framework for quantum error- correction, arXiv:2405.17567 10.48550/arXiv.2405.17567 (2024)

2024 doi arXiv
[16]

Combes, C

J. Combes, C. Ferrie, C. Cesare, M. Tiersch, G. J. Milburn, H. J. Briegel, and C. M. Caves, In-situ char- acterization of quantum devices with error correction, arXiv:1405.5656 10.48550/arXiv.1405.5656 (2014)

2014 doi arXiv
[17]

A. G. Fowler, D. Sank, J. Kelly, R. Barends, and J. M. Martinis, Scalable extraction of error models from the output of error detection circuits, arXiv:1405.1454 10.48550/arXiv.1405.1454 (2014)

2014 doi arXiv
[18]

Huo and Y

M.-X. Huo and Y. Li, Learning time-dependent noise to reduce logical errors: real time error rate estimation in quantum error correction, New Journal of Physics19, 123032 (2017)

2017
[19]

S. T. Spitz, B. Tarasinski, C. W. Beenakker, and T. E. O’Brien, Adaptive weight estimator for quantum error correction in a time-dependent environment, Advanced Quantum Technologies1, 1800012 (2018)

2018
[20]

Sivak, M

V. Sivak, M. Newman, and P. Klimov, Optimization of decoder priors for accurate quantum error correction, Physical Review Letters133, 150603 (2024)

2024
[21]

A. Remm, N. Lacroix, L. Bödeker, E. Genois, C. Hellings, F. Swiadek, G. J. Norris, C. Eichler, A. Blais, M. Müller, et al., Experimentally informed decoding of stabilizer codes based on syndrome correlations, Physical Review Research8, 013044 (2026)

2026
[22]

H. Wang, P. Liu, Y. Liu, J. Gu, J. Baker, F. T. Chong, and S. Han, Dgr: Tackling drifted and correlated noise in quantum error correction via decoding graph re- weighting, arXiv:2311.16214 10.48550/arXiv.2311.16214 (2023)

2023 doi arXiv
[23]

Gicev, L

S. Gicev, L. C. Hollenberg, and M. Usman, Quantum computer error structure probed by quantum error cor- rection syndrome measurements, Physical Review Re- search6, 043249 (2024)

2024
[24]

Bhardwaj, E

D. Bhardwaj, E. Takou, Y. Lin, and K. R. Brown, Adaptive estimation of drifting noise in quantum error correction, arXiv:2511.09491 10.48550/arXiv.2511.09491 (2025)

2025 doi
[25]

Florjanczyk and T

J. Florjanczyk and T. A. Brun, In-situ adaptive en- coding for asymmetric quantum error correcting codes, arXiv:1612.05823 10.48550/arXiv.1612.05823 (2016)

2016 doi arXiv
[26]

A. B. Magann, N. E. Miller, R. Blume-Kohout, P. Maunz, and K. C. Young, Fast-feedback protocols for calibration and drift control in quantum computers, arXiv:2512.07815 10.48550/arXiv.2512.07815 (2025)

2025 doi
[27]

Sivak, A

V. Sivak, A. Morvan, M. Broughton, R. G. Cortiñas, J. Bausch, A. W. Senior, M. Neeley, A. Eickbusch, N. Shutty, L. A. Beni,et al., Reinforcement learning con- trol of quantum error correction, Nature655, 879 (2026)

2026
[28]

Guatto, F

M. Guatto, F. Preti, M. Schilling, T. Calarco, F. A. Cárdenas-López, and F. Motzoi, Real-time adaptive quantum error correction by model-free multi-agent learning, arXiv:2509.03974 10.48550/arXiv.2509.03974 (2025)

2025 doi arXiv
[29]

Dennis, A

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics43, 4452 (2002)

2002
[30]

M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information(Cambridge university press, 2010)

2010
[31]

Tomita and K

Y. Tomita and K. M. Svore, Low-distance surface codes under realistic quantum noise, Physical Review A90, 062320 (2014)

2014
[32]

Krantz, M

P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gus- tavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Applied physics reviews6, 10.1063/1.5089550 (2019)

2019 doi
[33]

A. G. Fowler, Analytic asymptotic performance of topo- logical codes, Physical Review A—Atomic, Molecular, and Optical Physics87, 040301 (2013)

2013
[34]

Bravyi and A

S. Bravyi and A. Vargo, Simulation of rare events in quantum error correction, Physical Review A88, 062308 (2013)

2013
[35]

Bravyi, A

S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low- overhead fault-tolerant quantum memory, Nature627, 778 (2024)

2024
[36]

Bluvstein, A

D. Bluvstein, A. A. Geim, S. H. Li, S. J. Evered, J. P. Bonilla Ataides, G. Baranes, A. Gu, T. Manovitz, M. Xu, M. Kalinowski,et al., A fault-tolerant neutral-atom ar- chitecture for universal quantum computation, Nature 649, 39 (2026)

2026
[37]

Higgott, Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching, ACM Transactions on Quantum Computing3, 1 (2022)

O. Higgott, Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching, ACM Transactions on Quantum Computing3, 1 (2022)

2022
[38]

C. Wang, J. Harrington, and J. Preskill, Confinement- higgs transition in a disordered gauge theory and the ac- curacy threshold for quantum memory, Annals of Physics 303, 31 (2003)

2003
[39]

Wagner, H

T. Wagner, H. Kampermann, D. Bruß, and M. Kli- esch, Optimal noise estimation from syndrome statistics of quantum codes, Physical review research3, 013292 (2021)

2021
[40]

Wagner, H

T. Wagner, H. Kampermann, D. Bruß, and M. Kliesch, Learning logical pauli noise in quantum error correction, Physical review letters130, 200601 (2023)

2023
[41]

Hesner, B

I. Hesner, B. Hetényi, and J. R. Wootton, Using detector likelihood for benchmarking quantum error correction, Physical Review A111, 052452 (2025)

2025
[42]

Wang, C.-Y

P. Wang, C.-Y. Luan, M. Qiao, M. Um, J. Zhang, Y. Wang, X. Yuan, M. Gu, J. Zhang, and K. Kim, Single ion qubit with estimated coherence time exceeding one hour, Nature communications12, 233 (2021)

2021
[43]

Paetznick, B

A. Paetznick, B. Reichardt, M. da Silva, C. Ryan- Anderson, D. Aasen, J. Bello-Rivas, J. Campora, R. Chao, A. Chernoguzov, W. van Dam,et al., Improved quantum processor logical error rates via correction and detection, Nature654, 349 (2026)

2026
[44]

Berritta, J

F. Berritta, J. Benestad, J. A. Krzywda, O. Krause, M. A. Marciniak, S. Krøjer, C. W. Warren, E. Hogedal, A. Nylander, I. Ahmad,et al., Real-time adaptive track- ing of fluctuating relaxation rates in superconducting qubits, Physical Review X16, 011025 (2026)

2026
[45]

Arvind, N

R. Arvind, N. Bansal, D. E. Koh, T. Haug, and K. Bharti, Quantum error correction in adversar- ial regimes,arXiv:2509.08943 10.48550/arXiv.2509.08943 7 (2025)

2025 doi arXiv
[46]

Amezcua, L

M. Amezcua, L. Norris, T. Gilliss, R. Sitler, J. Shack- ford, G. Quiroz, and K. Schultz, Assessing spatiotem- porally correlated noise in superconducting qubits via pulse-based quantum noise spectroscopy, arXiv preprint arXiv:2603.19373 10.48550/arXiv.2603.19373 (2026)

2026 doi
[47]

A. M. Stephens, Z. W. Evans, S. J. Devitt, and L. C. Hol- lenberg, Asymmetric quantum error correction via code conversion, Physical Review A—Atomic, Molecular, and Optical Physics77, 062335 (2008)

2008
[48]

C. D. Hill, A. G. Fowler, D. S. Wang, and L. C. Hol- lenberg, Fault-tolerant quantum error correction code conversion, arXiv:1112.2417 10.48550/arXiv.1112.2417 (2011). 8 Supplemental Material In the Supplemental Material, we provide further proofs underlying our main results a...

2011 doi arXiv
[49]

Limit of small burst noise ratios 13
[50]

Detector-likelihood burst controller 15

Optimal burst noise advantage 14 F. Detector-likelihood burst controller 15
[51]

Numerical verification of optimal syndrome interval 17 H

Explicit bound 16 G. Numerical verification of optimal syndrome interval 17 H. Mapping to Google Willow experiment 17 I. Effective noise suppression via optimized intervals 20
[52]

Distance reduction via optimized intervals 21 J. Coherence-time estimate of the idle contribution 21 Appendix A: Phenomenological noise simulation of rotated surface code In the main text, we simulate a rotated surface-code memory experiment against bit-flip noise. We sample p...
[53]

Letr= 1 +ϵwithϵ≪1, and letL∈ {1, r}denote the instantaneous noise multiplier, withL= 1for a fraction1−fof the time andL=rfor a fractionf

Limit of small burst noise ratios Now, let us consider the limit of small burst noise ratiosr−1. Letr= 1 +ϵwithϵ≪1, and letL∈ {1, r}denote the instantaneous noise multiplier, withL= 1for a fraction1−fof the time andL=rfor a fractionf. Then µ≡E[L] = (1−f) +f r, Var(L) µ2 = f(1−...
[54]

For this, we optimize Eq

Optimal burst noise advantage Next, we leave the smallrlimit and find the optimalrover all burst duration fractionsf. For this, we optimize Eq. (E7) over the burst duration fractionf, i.e. searching for the burst duration which gives the highest advantage. For Eq. (E7), thez=z...
[55]

The purpose is to show that a burst can be identified from only a small number of syndrome rounds when the code distance is large

Explicit bound We now give an explicit finite-sample bound. The purpose is to show that a burst can be identified from only a small number of syndrome rounds when the code distance is large. Let us assume we start out with the no-burst interval∆tlow. Then, we define the detect...
[56]

what isd opt that gives the same logical error as a fixed-timing code of distancedfixed

Distance reduction via optimized intervals Next, we ask how much distance can one save by using the optimized interval while preserving the logical error, i.e. what isd opt that gives the same logical error as a fixed-timing code of distancedfixed. This condition is Ropt(dopt)...

Pith tools

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

[1] [1]

D. A. Lidar and T. A. Brun,Quantum error correction (Cambridge university press, 2013)

2013

[2] [2]

B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics87, 307 (2015)

2015

[3] [3]

D. J. Spencer, S. P. Jain, A. Tanggara, Z. Sun, T. Haug, D. Khu, and K. Bharti, Quantum error cor- rection and fault tolerance: A comprehensive tutorial, arXiv:2605.29137 10.48550/arXiv.2605.29137 (2026)

work page Pith review arXiv doi:10.48550/arxiv.2605.29137 2026

[4] [4]

C. A. Pattison, M. E. Beverland, M. P. da Silva, and N. Delfosse, Improved quantum error correction us- ing soft information, arXiv preprint arXiv:2107.13589 10.48550/arXiv.2107.13589 (2021)

arXiv doi:10.48550/arxiv.2107.13589 2021

[5] [5]

Márton and J

Á. Márton and J. K. Asbóth, Optimal number of stabi- lizer measurement rounds in an idling surface code patch, Quantum9, 1767 (2025)

work page 2025

[6] [6]

Maurya and S

S. Maurya and S. Tannu, Synchronization for fault- tolerant quantum computers, inProceedings of the 52nd Annual International Symposium on Computer Architec- ture(2025) pp. 1370–1385

work page 2025

[7] [7]

P. S. Mundada, A. Khindanov, Y. Wang, C. L. Ed- munds, P. Coote, M. J. Biercuk, Y. Baum, and M. Hush, Heterogeneous architectures enable a 138x reduction in physical qubit requirements for fault-tolerant quantum computing under detailed accounting, arXiv preprint arXiv:2604.06319 10.48550/arXiv.2604.06319 (2026)

arXiv doi:10.48550/arxiv.2604.06319 2026

[8] [8]

S. M. Girvin, Introduction to quantum error correction and fault tolerance, SciPost Physics Lecture Notes , 070 (2023)

work page 2023