Diagnosing and Restoring the Degraded Fault Distance of Magic State Cultivation

Abstract

T-state cultivation is a resource-efficient protocol producing logical T states but recent benchmarks show that its logical error rate is considerably higher than intended i.e. than S-state cultivation, which is the analogous protocol for producing logical S states. In this paper, we explain this T-S discrepancy by analytically showing, under circuit-level depolarising noise, that distance-3 (-5) T-state cultivation has fault distance 2 (3) due to Pauli hook errors that propagate to coherent Clifford errors after its final double-check circuit; such errors remain Pauli in S-state cultivation, which consequently retains fault distance 3 (5). As part of our analysis we derive a general formula, and an 𝒪(n³)-time algorithm for fixed logical-qubit count, for the acceptance probability of a logical mixed state afflicted with a Clifford error, where n is the physical qubit count. We then design flags that detect the malignant hook errors in cultivation, improving the pre-escape logical error rate from Θ(p³) to Θ(p⁵). At noise level p =10⁻³, this is a 4.9× improvement, costing a 1.36× increase in attempts per accepted shot.

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BibTeX RIS

BibTeX

@misc{chan2026diagnosing,
  title = {{Diagnosing and Restoring the Degraded Fault Distance of Magic State Cultivation}},
  author = {Tim Chan and Armands Strikis and Zhu Sun and Zhenyu Cai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17706},
  eprint = {2609.17706},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17706},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Diagnosing and Restoring the Degraded Fault Distance of Magic State Cultivation
AU  - Tim Chan
AU  - Armands Strikis
AU  - Zhu Sun
AU  - Zhenyu Cai
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.17706
UR  - https://arxiv.org/abs/2609.17706
AB  - T-state cultivation is a resource-efficient protocol producing logical T states but recent benchmarks show that its logical error rate is considerably higher than intended i.e. than S-state cultivation, which is the analogous protocol for producing logical S states. In this paper, we explain this T-S discrepancy by analytically showing, under circuit-level depolarising noise, that distance-3 (-5) T-state cultivation has fault distance 2 (3) due to Pauli hook errors that propagate to coherent Clifford errors after its final double-check circuit; such errors remain Pauli in S-state cultivation, which consequently retains fault distance 3 (5). As part of our analysis we derive a general formula, and an 𝒪(n^3)-time algorithm for fixed logical-qubit count, for the acceptance probability of a logical mixed state afflicted with a Clifford error, where n is the physical qubit count. We then design flags that detect the malignant hook errors in cultivation, improving the pre-escape logical error rate from Θ(p^3) to Θ(p^5). At noise level p =10^−3, this is a 4.9× improvement, costing a 1.36× increase in attempts per accepted shot.
ER  -

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