Exact logical error rates for magic state cultivation
Abstract
We compute exactly the acceptance and logical error rates for the distance d=3 and d=5 magic state cultivation circuits from Clifft [arXiv:2604.27058] and SOFT [arXiv:2512.23037] using Pauli propagation and binary tensor contraction. Actual T-gates are studied, not the S-gate proxy used for sampling. The calculation includes every fault order at several circuit-level noise strengths (p). We provide a series expansion form to the logical error rates, through order (p/(1−p))10. The analytical results recover the numerical values from Clifft and SOFT at both d=3 and d=5 to within their sampling uncertainty. Then, we show that the d=3 and d=5 circuits actually have fault distances of dfault=2 and dfault=3 respectively, explaining the similar distance degrading effects from the companion code of [Quantum 10, 2134 (2026)].
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BibTeX
@misc{wan2026exact,
title = {{Exact logical error rates for magic state cultivation}},
author = {Kwok Ho Wan and Ainhoa Zapirain},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18922},
eprint = {2609.18922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18922},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact logical error rates for magic state cultivation
AU - Kwok Ho Wan
AU - Ainhoa Zapirain
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.18922
UR - https://arxiv.org/abs/2609.18922
AB - We compute exactly the acceptance and logical error rates for the distance d=3 and d=5 magic state cultivation circuits from Clifft [arXiv:2604.27058] and SOFT [arXiv:2512.23037] using Pauli propagation and binary tensor contraction. Actual T-gates are studied, not the S-gate proxy used for sampling. The calculation includes every fault order at several circuit-level noise strengths (p). We provide a series expansion form to the logical error rates, through order (p/(1−p))^10. The analytical results recover the numerical values from Clifft and SOFT at both d=3 and d=5 to within their sampling uncertainty. Then, we show that the d=3 and d=5 circuits actually have fault distances of d_fault=2 and d_fault=3 respectively, explaining the similar distance degrading effects from the companion code of [Quantum 10, 2134 (2026)].
ER -