arxiv: 2605.05881 · v1 · submitted 2026-05-07 · 🪐 quant-ph

Recognition: unknown

Surface-Code Thresholds and Qubit Footprints in Shuttling-Based Spin-Qubit Railways

Arun John Moncy , Reza Dastbasteh , Josu Etxezarreta Martinez , Ryo Nagai , Pedro M. Crespo , Normann Mertig , Charles Smith , Ruben M. Otxoa

Authors on Pith no claims yet

Pith reviewed 2026-05-08 11:34 UTC · model grok-4.3

classification 🪐 quant-ph

keywords surface codesspin qubitselectron shuttlingfault tolerancequantum error correctionsilicon spin qubitsXZZX codeMegaquop

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The pith

Tailoring XZZX surface codes to dephasing bias from shuttling allows a distance-7 code to reach Megaquop performance at physical error rates of 10^{-3} in 2xN spin-qubit railways.

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

The paper maps rotated surface codes onto a linear silicon spin-qubit array that uses electron shuttling along a 2xN railway to bypass wiring fan-out limits. It finds that moving check qubits rather than data qubits raises the overall error threshold, and that an XZZX non-CSS code beats standard CSS codes when the dominant shuttling noise is dephasing. With this bias-matched choice, a distance-7 code already delivers the Megaquop regime at a physical error rate of 10^{-3}, cutting the qubit count needed for early fault-tolerant processors.

Core claim

We present a fault-tolerant mapping of rotated surface codes onto a 2×N silicon spin-qubit railway architecture, utilizing electron shuttling to resolve the wiring fan-out bottleneck. Employing circuit-level noise modeling, we evaluate threshold performances across various noise biases. We demonstrate that shuttling check qubits instead of data qubits fundamentally improves system thresholds. Crucially, under a noise model biased towards dephasing for spin-qubit shuttling, the non-CSS XZZX surface code outperforms standard CSS variants. By tailoring the topological code to this specific inherent bias, we show that the Megaquop footprint is achievable with a distance 7 code requiring a p = 10

What carries the argument

The XZZX surface code adapted to dephasing-biased shuttling noise in a 2xN spin-qubit railway, where check-qubit shuttling raises the threshold and shrinks the logical-qubit footprint.

If this is right

Shuttling check qubits rather than data qubits raises the fault-tolerance threshold under the same physical noise.
The XZZX code outperforms CSS codes when shuttling noise is dominated by dephasing.
A distance-7 code suffices to reach the Megaquop regime at a physical error rate of 10^{-3}.
The approach yields a concrete reduction in total physical qubits needed for early fault-tolerant operation.

Where Pith is reading between the lines

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

The same bias-matching strategy might extend to other platforms that move qubits or use shuttling-like operations under dephasing-dominant noise.
Lowering the required code distance could shorten the path to demonstrating small-scale fault-tolerant algorithms on near-term hardware.
Further refinements to shuttling timing or trajectory might push the effective error rate below 10^{-3} without changing the code distance.

Load-bearing premise

Shuttling check qubits introduces no connectivity or timing errors beyond the dephasing captured in the circuit-level noise model, and that this dephasing accurately represents real-device shuttling noise.

What would settle it

A direct measurement in a silicon spin-qubit device of the error rate and bias when shuttling check qubits versus data qubits, to test whether the modeled dephasing threshold and XZZX advantage appear at p around 10^{-3}.

Figures

Figures reproduced from arXiv: 2605.05881 by Arun John Moncy, Charles Smith, Josu Etxezarreta Martinez, Normann Mertig, Pedro M. Crespo, Reza Dastbasteh, Ruben M. Otxoa, Ryo Nagai.

**Figure 1.** Figure 1: FIG. 1. Schematic representation mapping the 2D rotated view at source ↗

**Figure 3.** Figure 3: FIG. 3. Syndrome extraction circuit for an view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 4.** Figure 4: FIG. 4. Three single qubit errors: two hook errors and a view at source ↗

**Figure 5.** Figure 5: FIG. 5. The extraction order to avoid hook errors aligning view at source ↗

**Figure 7.** Figure 7: FIG. 7. RSC code with syndrome extraction orders chosen view at source ↗

**Figure 8.** Figure 8: FIG. 8. Data qubit order satisfying the order rules in Fig: view at source ↗

**Figure 10.** Figure 10: FIG. 10. Map of Syndrome extraction for d=5 RSC in Train view at source ↗

**Figure 9.** Figure 9: FIG. 9. Data and check qubit order ( view at source ↗

**Figure 11.** Figure 11: FIG. 11. The logical error rates per round versus the ro view at source ↗

read the original abstract

We present a fault-tolerant mapping of rotated surface codes onto a $2\times N$ silicon spin-qubit railway architecture, utilizing electron shuttling to resolve the wiring fan-out bottleneck. Employing circuit-level noise modeling, we evaluate threshold performances across various noise biases. We demonstrate that shuttling check qubits instead of data qubits fundamentally improves system thresholds. Crucially, under a noise model biased towards dephasing for spin-qubit shuttling, the non-CSS XZZX surface code outperforms standard CSS variants. By tailoring the topological code to this specific inherent bias, we show that the Megaquop footprint is achievable with a distance 7 code requiring a $p = 10^{-3}$ physical error rate, highlighting a pathway for substantial hardware reductions in early fault-tolerant quantum processors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Shuttling check qubits on a 2xN spin-qubit rail plus the XZZX code yields a d=7 Megaquop path at p=10^{-3} under dephasing bias, but the noise model details are too thin to verify the scaling.

read the letter

The paper maps rotated surface codes onto a shuttling-based 2xN silicon spin-qubit railway and reports that moving the check qubits instead of the data qubits raises the threshold. Under a dephasing-biased noise model from shuttling, the XZZX variant beats standard CSS codes, and they claim a distance-7 code reaches the Megaquop regime at a physical error rate of 10^{-3}. That combination of layout and code choice is the concrete new piece for this hardware platform. The circuit-level Monte Carlo approach is the right tool for the job and avoids obvious circularity in the thresholds. The architecture-specific tailoring is a legitimate extension of existing surface-code work. The soft spot is the missing detail on how shuttling errors are actually implemented. The headline result only holds if shuttling the checks adds no extra correlated noise, timing jitter, or position-dependent effects beyond the simple dephasing channel. The abstract and available description leave that assumption untested in the visible material, which is the main verification gap. This is for groups working on spin-qubit hardware who need concrete footprint numbers for early fault-tolerant planning. It is worth a serious referee who can ask for the full noise model, simulation code, and robustness checks on the shuttling assumptions. I would send it to peer review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a fault-tolerant mapping of rotated surface codes (including the non-CSS XZZX variant) onto a 2×N silicon spin-qubit railway architecture that uses electron shuttling to mitigate wiring fan-out. Circuit-level Monte Carlo simulations under biased Pauli noise models are used to evaluate thresholds; the central claim is that shuttling check qubits (rather than data qubits) improves performance, and that under a dephasing-biased noise model the XZZX code reaches a Megaquop logical footprint with a distance-7 code at physical error rate p = 10^{-3}.

Significance. If the underlying noise-model assumptions hold, the work identifies a concrete route to substantially smaller qubit counts for early fault-tolerant processors by co-designing the code with the shuttling-induced dephasing bias. This is a practically relevant direction for spin-qubit hardware.

major comments (2)

[Circuit-level noise modeling and shuttling implementation] The headline result (d = 7 XZZX reaching Megaquop footprint at p = 10^{-3}) rests on the modeling choice that shuttling check qubits introduces only independent, dephasing-biased Pauli errors fully captured by the circuit-level noise channel. No explicit validation or parameter sweep is provided showing that position-dependent correlations, shuttle-induced timing jitter, or residual exchange interactions remain negligible in the 2×N railway geometry; this assumption is load-bearing for both the threshold improvement and the claimed hardware reduction.
[Numerical results and threshold extraction] The reported logical-error scaling and footprint numbers for the XZZX code are obtained from direct Monte Carlo simulation; the manuscript does not supply the precise shuttling-error implementation details, data-exclusion rules, or convergence diagnostics that would allow independent reproduction of the d = 7 threshold curve.

minor comments (2)

[Abstract and introduction] Define or cite the precise definition of 'Megaquop footprint' (e.g., logical error rate per logical qubit per cycle) when first used in the abstract and introduction.
[Figures presenting threshold data] Figure captions or legends should list the exact bias ratios (e.g., dephasing-to-depolarizing ratio) and the number of Monte Carlo shots used for each threshold point.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript regarding shuttling-based spin-qubit railways and surface-code thresholds. We address each major comment point by point below, clarifying our modeling assumptions and committing to improvements for reproducibility where appropriate.

read point-by-point responses

Referee: [Circuit-level noise modeling and shuttling implementation] The headline result (d = 7 XZZX reaching Megaquop footprint at p = 10^{-3}) rests on the modeling choice that shuttling check qubits introduces only independent, dephasing-biased Pauli errors fully captured by the circuit-level noise channel. No explicit validation or parameter sweep is provided showing that position-dependent correlations, shuttle-induced timing jitter, or residual exchange interactions remain negligible in the 2×N railway geometry; this assumption is load-bearing for both the threshold improvement and the claimed hardware reduction.

Authors: We agree that the independent dephasing-biased Pauli error model is a key assumption. In the manuscript we motivate shuttling only check qubits precisely to minimize data-qubit exposure to shuttle-induced effects, and we cite prior spin-qubit shuttling studies indicating that residual exchange and jitter can be suppressed below the 10^{-3} level in linear geometries. However, we did not include an explicit parameter sweep or correlation analysis. In revision we will add a dedicated paragraph in the methods section discussing the expected magnitude of these effects in the 2×N railway (drawing on existing device literature) and a brief sensitivity test showing that moderate correlations do not qualitatively alter the reported threshold ordering. We cannot perform a full device-specific validation without additional experimental input, but the added discussion will make the load-bearing nature of the assumption transparent. revision: partial
Referee: [Numerical results and threshold extraction] The reported logical-error scaling and footprint numbers for the XZZX code are obtained from direct Monte Carlo simulation; the manuscript does not supply the precise shuttling-error implementation details, data-exclusion rules, or convergence diagnostics that would allow independent reproduction of the d = 7 threshold curve.

Authors: We accept that the current manuscript lacks sufficient implementation specifics for full reproducibility. In the revised version we will add an appendix containing: (i) the exact circuit-level error channel applied during shuttling operations, (ii) the data-exclusion criteria used (e.g., discarding Monte Carlo runs that exceed a maximum syndrome weight), and (iii) convergence diagnostics including sample counts, statistical error bars, and the number of logical errors observed for the d = 7 XZZX curve at p = 10^{-3}. These additions will enable independent verification of the reported Megaquop footprint. revision: yes

Circularity Check

0 steps flagged

No circularity detected; thresholds obtained from direct circuit-level Monte Carlo simulations

full rationale

The paper's central results consist of threshold values and footprint estimates computed via numerical Monte Carlo sampling of rotated surface codes under a circuit-level dephasing-biased Pauli noise model. These quantities are not obtained by fitting parameters to the target Megaquop footprint, nor are they defined in terms of the final claim. No self-definitional equations, fitted-input predictions, or load-bearing self-citations appear in the derivation; the mapping of XZZX codes onto the 2xN railway and the shuttling-check-qubit choice are modeling decisions whose consequences are evaluated externally by simulation rather than by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard quantum error correction assumptions plus domain-specific modeling choices for shuttling noise that are not independently validated in the abstract.

free parameters (2)

physical error rate p
The value p = 10^{-3} is the operating point at which the distance-7 megaquop footprint is claimed.
noise bias parameters
Multiple bias ratios are swept in the circuit-level simulations to identify the dephasing-favoring regime.

axioms (2)

domain assumption Circuit-level noise model accurately captures all relevant errors during shuttling operations.
Invoked to evaluate thresholds across biases.
domain assumption Shuttling check qubits incurs no additional uncontrolled connectivity or timing errors beyond the modeled noise.
Central to the claim that shuttling check qubits improves thresholds.

pith-pipeline@v0.9.0 · 5468 in / 1575 out tokens · 56232 ms · 2026-05-08T11:34:35.556117+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

56 extracted references · 20 canonical work pages

[1]

check qubit

versus moving check qubits around stationary data [15], demonstrating that moving the check qubits sys- tematically yields to high fault-tolerant thresholds. We continue our study by numerically exploring the mega- and gigaquop footprints for two experimentally relevant physical error rates, p = 0.003 and p = 0.001. We observe that as a result of the bias...

2047
[2]

R. Li, L. Petit, D. Franke, J. Dehollain, J. Helsen, M. Steudtner, N. Thomas, Z. Yoscovits, K. Singh, S. Weperen,et al., Science advances4, eaar3960 (2018)

2018
[3]

D. P. Franke, J. S. Clarke, L. M. Vandersypen, and M. Veldhorst, Microprocessors and Microsystems67, 1 (2019)

2019
[4]

M. F. Gonzalez-Zalba, S. de Franceschi, E. Charbon, T. Meunier, M. Vinet, and A. S. Dzurak, Nature Elec- tronics4, 872 (2021)

2021
[5]

12/15 Figure 6.Damping rates of Rabi oscillations near the anti-crossing

S. Kubiceket al., arXiv preprint arXiv:2409.12731 (2024), 10.1038/s41534-025-01016-x

work page doi:10.1038/s41534-025-01016-x 2024
[6]

T. Koch, C. Godfrin, V. Adam,et al., npj Quantum In- formation11, 59 (2025)

2025
[7]

Veldhorstet al., Nature Communications8, 1766 (2017)

M. Veldhorstet al., Nature Communications8, 1766 (2017)

2017
[8]

Liet al., Science Advances4, eaar3960 (2018)

R. Liet al., Science Advances4, eaar3960 (2018)

2018
[9]

Brousseet al., arXiv preprint (2022), arXiv:2208.08333

D. Brousseet al., arXiv preprint (2022), arXiv:2208.08333

work page arXiv 2022
[10]

Yoneda, W

J. Yoneda, W. Huang, M. Feng, C. Yang, K. Chan, T. Tanttu, W. Gilbert, R. Leon, F. Hudson, K. Itoh, et al., Nature communications12, 3792 (2021)

2021
[11]

Millset al., Nature Communications10, 1063 (2019)

A. Millset al., Nature Communications10, 1063 (2019)

2019
[12]

Zwerveret al., PRX Quantum4, 030303 (2023)

A. Zwerveret al., PRX Quantum4, 030303 (2023)

2023
[13]

De Smetet al., arXiv preprint arXiv:2406.07267 (2024), 10.1038/s41565-025-01920-5

M. De Smetet al., arXiv preprint arXiv:2406.07267 (2024), 10.1038/s41565-025-01920-5

work page doi:10.1038/s41565-025-01920-5 2024
[14]

Volmeret al., npj Quantum Information10, 61 (2024)

M. Volmeret al., npj Quantum Information10, 61 (2024)

2024
[15]

Strucket al., Nature Communications15, 1325 (2024)

T. Strucket al., Nature Communications15, 1325 (2024). 14

2024
[16]

and Bhardwaj, Harsh and Beer, Max and Kammerloher, Eugen and Otten, René and Seidler, Inga and Xue, Ran and Schreiber, Lars R

M. K¨ unne, A. Willmes, M. Oberl¨ ander, C. Gorjaew, J. D. Teske, H. Bhardwaj, M. Beer, E. Kammerloher, R. Ot- ten, I. Seidler,et al., arXiv preprint arXiv:2306.16348 (2023), 10.1038/s41467-024-49182-4

work page doi:10.1038/s41467-024-49182-4 2023
[17]

Ginzelet al., Physical Review B110, 075302 (2024)

F. Ginzelet al., Physical Review B110, 075302 (2024)

2024
[18]

S. M. Patom¨ aki, M. F. Gonzalez-Zalba, M. A. Fogarty, Z. Cai, S. C. Benjamin, and J. J. L. Morton, arXiv preprint arXiv:2306.07673 (2023), 10.1038/s41534-024- 00823-y

work page doi:10.1038/s41534-024- 2023
[19]

R. M. Otxoa, J. Etxezarreta Martinez, P. Schnabl, N. Mertig, C. Smith, and F. Martins, arXiv preprint arXiv:2504.03149 (2025), 10.1007/978-3-031-78219-0 12

work page doi:10.1007/978-3-031-78219-0 2025
[20]

Ginzel, A

F. Ginzel, A. R. Mills, J. R. Petta, and G. Burkard, Phys. Rev. B102, 195418 (2020)

2020
[21]

D. W. Kanaar, U. G¨ ung¨ ord¨ u, and J. P. Kestner, Philos. Trans. R. Soc. A380, 20210275 (2022)

2022
[22]

R. Li, V. Levajac, and C. e. Godfrin, arXiv preprint arXiv:2501.17814 (2025), 10.21236/ada623828

work page doi:10.21236/ada623828 2025
[23]

Escofet, E

P. Escofet, E. Alarc´ on, S. Abadal, A. Semenov, N. Mur- phy, E. Blokhina, and C. G. Almud´ ever, arXiv preprint arXiv:2510.17689 (2025), 10.1103/4p4q-fm3y

work page doi:10.1103/4p4q-fm3y 2025
[24]

J. D. Chadwick, W. Yang, J. Viszlai, and F. T. Chong, arXiv preprint (2025), arXiv:2512.07131

work page arXiv 2025
[25]

Tokura, W

Y. Tokura, W. G. Van Der Wiel, T. Obata, and S. Tarucha, Physical review letters96, 047202 (2006)

2006
[26]

Pioro-Ladri` ere, T

M. Pioro-Ladri` ere, T. Obata, Y. Tokura, Y.-S. Shin, T. Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Nature Physics4, 776 (2008)

2008
[27]

Yoneda, T

J. Yoneda, T. Otsuka, T. Nakajima, T. Takakura, T. Obata, M. Pioro-Ladri` ere, H. Lu, C. J. Palmstrøm, A. C. Gossard, and S. Tarucha, Applied Physics Express 8, 084401 (2015)

2015
[28]

Klemtet al., npj Quantum Information9, 107 (2023)

B. Klemtet al., npj Quantum Information9, 107 (2023)

2023
[29]

Greschet al., Phys

D. Greschet al., Phys. Rev. B104, L161303 (2021)

2021
[30]

Bosco, J

S. Bosco, J. Zou, and D. Loss, arXiv preprint (2023), arXiv:2311.15970

work page arXiv 2023
[31]

Ademiet al., arXiv preprint (2025), arXiv:2510.26860

Z. Ademiet al., arXiv preprint (2025), arXiv:2510.26860

work page arXiv 2025
[32]

Siegel, G

S. Siegel, G. Strikis, and M. Fogarty, PRX Quantum5, 040328 (2024)

2024
[33]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Phys. Rev. A86, 032324 (2012)

2012
[34]

Litinski, Quantum3, 128 (2019)

D. Litinski, Quantum3, 128 (2019)

2019
[35]

J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, Nature Communications12, 2172 (2021)

2021
[36]

Etxezarreta Martinez, P

J. Etxezarreta Martinez, P. Schnabl, J. Oliva del Moral, R. Dastbasteh, P. M. Crespo, and R. M. Otxoa, Phys. Rev. Applied25, 014021 (2026)

2026
[37]

Tomita and K

Y. Tomita and K. M. Svore, Physical Review A89, 032306 (2014)

2014
[38]

Spitz, B

S. Spitz, B. Taranto,et al., arXiv preprint arXiv:2107.14252 (2021), 10.22331/q-2022-09-19-809

work page doi:10.22331/q-2022-09-19-809 2021
[39]

deMarti iOlius, P

A. deMarti iOlius, P. Fuentes, R. Or´ us, P. M. Crespo, and J. Etxezarreta Martinez, Quantum8, 1498 (2024)

2024
[40]

Generalized bicycle codes with low con- nectivity: Minimum distance bounds and hook errors,

R. Dastbasteh, O. Sanz Larrarte, A. J. Moncy, P. M. Crespo, J. Etxezarreta Martinez, and R. M. Otxoa, “Generalized bicycle codes with low connectivity: Min- imum distance bounds and hook errors,” (2025), arXiv:2508.09082 [cs.IT]

work page arXiv 2025
[41]

Gidney, Quantum5, 497 (2021)

C. Gidney, Quantum5, 497 (2021)

2021
[42]

Higgott and C

O. Higgott and C. Gidney, Quantum9, 1600 (2025)

2025
[43]

J. Y. Huanget al., Nature627, 772 (2024)

2024
[44]

N. D. Stuycket al., (2024), 10.1109/snw63608.2024.10639218

work page doi:10.1109/snw63608.2024.10639218 2024
[45]

S. G. J. Philipset al., Nature609, 919 (2022)

2022
[46]

Wanget al., Science385, 447 (2024)

C.-A. Wanget al., Science385, 447 (2024)

2024
[47]

Bassiet al., (2024), arXiv:2412.13069

M. Bassiet al., (2024), arXiv:2412.13069

work page arXiv 2024
[48]

Presentstatusand future challenges of non-interferometric tests of collapse models.Nature Phys., 18(3):243–250, 2022

T. Tanttuet al., Nature Physics (2024), 10.1038/s41567- 024-02614-w

work page doi:10.1038/s41567- 2024
[49]

Xueet al., Nature601, 343 (2022)

X. Xueet al., Nature601, 343 (2022)

2022
[50]

Geyeret al., Nature Physics20(2024), 10.1038/s41567-024-02481-5

S. Geyeret al., Nature Physics20(2024), 10.1038/s41567-024-02481-5

work page doi:10.1038/s41567-024-02481-5 2024
[51]

G. A. Oakeset al., Phys. Rev. X13, 011023 (2023)

2023
[52]

Takedaet al., npj Quantum Information10(2024), 10.1038/s41534-024-00813-0

K. Takedaet al., npj Quantum Information10(2024), 10.1038/s41534-024-00813-0

work page doi:10.1038/s41534-024-00813-0 2024
[53]

van Riggelen-Doelmanet al., Nature Communications 15, 5716 (2024)

F. van Riggelen-Doelmanet al., Nature Communications 15, 5716 (2024)

2024
[54]

N. W. Hendrickxet al., Nature591(2021), 10.1038/s41586-021-03332-6

work page doi:10.1038/s41586-021-03332-6 2021
[55]

L. C. Camenzindet al., Nature Electronics (2022), 10.1038/s41928-022-00722-0

work page doi:10.1038/s41928-022-00722-0 2022
[56]

A. R. O’Rourke and S. Devitt, Phys. Rev. Res.7, 033074 (2025)

2025