Good quantum codes with addressable and parallelizable non-Clifford gates

Abstract: 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.