发表机构
University of California San Diego（加利福尼亚大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了具有横向A7对称性的双参数七四元纠错码族，通过不变父态方法实现非克利福德操作，并揭示了连续自由度来源。
AI 中文摘要
我们构造了一个双参数族的单错误纠正七四元码，具有横向的 $\tilde{A_7}$ 对称性，实现了双量子比特超黄金门集的有限分量。这些 $((7,4,3))_4$ 码编码两个逻辑量子比特，并通过将相同的门应用于每个物理四元来支持非克利福德操作。我们的方法构造不变父态，而不是直接搜索码空间。对于局部维度为 $q$ 的 $n$ 个系统，假设局部对称性是一个酉 $t$-群，置换对称性是 $t$-传递的。那么，所有最多 $t$ 个位点上的边缘的最大混合等价于 Schur--Weyl 扇区权重中的线性方程。每个可行解产生距离至少为 $t+1$ 的纯父态；穿刺一个位点产生一个距离至少为 $t$ 的 $q$ 维码，并具有规定的局部对称性横向作用。Steane 码和非稳定子 Fake Steane 码揭示了为什么构造留下连续自由度。它们共享相同的规定的横向克利福德和置换对称性，它们的八量子比特纯化位于一个圆上：纠错固定两个不变扇区范数，但留下相对相位自由。保持相同的置换群并将克利福德对称性替换为 $\tilde{A_7} \leq SU(4)$，得到三个固定的扇区范数和两个自由的相对相位，产生了新的四元码族。
英文摘要
We construct a two-parameter family of single-error-correcting seven-ququart codes with transversal $\tilde{A_7}$ symmetry, realizing the finite component of a two-qubit super-golden gate set. These $((7,4,3))_4$ codes encode two logical qubits and support non-Clifford operations by applying the same gate to each physical ququart. Our method constructs invariant parent states rather than searching directly for a codespace. For $n$ systems of local dimension $q$, suppose the local symmetry is a unitary $t$-group and the permutation symmetry is $t$-transitive. Maximal mixing of all marginals on at most $t$ sites is then equivalent to linear equations in Schur--Weyl sector weights. Every feasible solution yields parents of pure distance at least $t+1$; puncturing one site produces a $q$-dimensional code of distance at least $t$ with the prescribed local symmetry acting transversally. Steane and the nonstabilizer Fake Steane code reveal why the construction leaves continuous freedom. They share the same prescribed transversal Clifford and permutation symmetries, and their eight-qubit purifications lie on a circle: error correction fixes two invariant-sector norms but leaves the relative phase free. Keeping the same permutation group and replacing Clifford symmetry by $\tilde{A_7}$ $\leq SU(4)$ gives three fixed sector norms and two free relative phases, producing the new ququart family.
