发表机构
Imperial College London; University of Oxford（帝国理工学院; 牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究精确计算了d=3和d=5魔法态培育电路的逻辑错误率，揭示了其实际故障距离低于预期，解释了距离退化效应。
AI 中文摘要
我们利用泡利传播和二进制张量收缩，精确计算了来自Clifft [arXiv:2604.27058]和SOFT [arXiv:2512.23037]的距离d=3和d=5魔法态培育电路的接受率和逻辑错误率。我们研究的是实际的T门，而非用于采样的S门代理。该计算涵盖了多个电路级噪声强度(p)下的所有故障阶数。我们提供了逻辑错误率的级数展开形式，直至(p/(1-p))^10阶。解析结果在d=3和d=5两种情况下均与Clifft和SOFT的数值结果在其采样不确定度内吻合。然后，我们展示了d=3和d=5电路的实际故障距离分别为d_fault=2和d_fault=3，这解释了[Quantum 10, 2134 (2026)]的伴随代码中类似的距离退化效应。
英文摘要
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 $d_{\text{fault}}=2$ and $d_{\text{fault}}=3$ respectively, explaining the similar distance degrading effects from the companion code of [Quantum 10, 2134 (2026)].
Commentsv3 - expanded section on Pauli semiwebs and implementation available at: https://github.com/kh428/exact_ler_for_msc
