We revisit a family of good quantum error-correcting codes presented in He $\textit{et al.}$ (2025), and we show that various sets of addressable and transversal non-Clifford multi-control-$Z$ gates can be performed in parallel. The construction relies on the good classical codes of Stichtenoth (IEEE Trans. Inf. Theory, 2006), which were previously instantiated in He $\textit{et al.}$ (2025), to yield quantum CSS codes over which addressable logical $\mathsf{CCZ}$ gates can be performed at least one at a time. Here, we show that for any $m$, there exists a family of good quantum error-correcting codes over qudits for which logical $\mathsf{C}^{m}\mathsf{Z}$ gates can address specific logical qudits and be performed in parallel. This leads to a significant advantage in the depth overhead of multi-control-$Z$ circuits.

翻译：我们重新审视了He等人（2025年）提出的一族优良量子纠错码，并证明了多种可寻址且横向的非克利福德多控制Z门可以并行执行。该构造基于Stichtenoth（IEEE Trans. Inf. Theory, 2006）的优良经典码——该码先前已在He等人（2025年）的工作中被实例化——以产生量子CSS码，使得可寻址的逻辑CCZ门能够至少每次执行一个。本文中，我们证明对于任意m，存在一族定义在量子比特上的优良量子纠错码，其逻辑C^mZ门能够寻址特定的逻辑量子比特并实现并行执行。这为多控制Z电路的深度开销带来了显著优势。