发表机构
Duke Quantum Center, Duke University; Duke University（杜克大学杜克量子中心; 杜克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
利用射影几何和SAT求解器构造具有横向CCZ门、距离至少为3的三逻辑量子比特CSS码，并证明在n<39时不存在此类码，填补了码长范围的部分空白。
AI 中文摘要
我们构造了具有三个逻辑量子比特的CSS码，其逻辑CCZ门由横向物理$T/T^{\dagger}$门实现，且距离至少为3。这些码从由射影几何定义的8整除$X$-稳定子出发，这保证了$d_Z\geq 3$。然后，SAT求解器找到三个与横向CCZ兼容的$X$-逻辑算子。我们在从$n=48$到$n=496$的十三个块长度上构建码，其中$d_Z=3$或$4$，并用$[[48,3,3]]$码$Q_{48}$说明该方法。此外，我们证明了不存在具有三个逻辑量子比特、$Z$-距离至少为3且具有准横向CCZ门（带有对角Clifford校正的横向$T$）的CSS码，当$n<39$时。Jacinto等人的$[[47,3,3]]$码作为上界，留下$39\leq n\leq46$未解决。
英文摘要
We construct CSS codes with three logical qubits whose logical CCZ gate is implemented by transversal physical $T/T^{\dagger}$ gates and whose distance is at least $3$. These codes start with $8$-divisible $X$-stabilizers defined by projective geometry, which guarantees $d_Z\geq 3$. A SAT solver then finds three $X$-logical operators compatible with the transversal CCZ. We build codes at thirteen block lengths from $n=48$ to $n=496$, with $d_Z=3$ or $4$, and illustrate the method with the $[[48,3,3]]$ code $Q_{48}$. Additionally, we prove that no CSS code with three logical qubits, $Z$-distance at least $3$, and a quasi-transversal CCZ gate (transversal $T$ with a diagonal Clifford correction) exists below $n=39$. The $[[47,3,3]]$ code of Jacinto et al. serves as an upper bound, leaving $39\leq n\leq46$ open.
Comments40 pages, 3 figures. Code and data: https://github.com/jerrylvx/gate2code-public
