Good quantum codes with addressable and parallelizable transversal non-Clifford gates

In this work, we prove that for any $m>1$, there exists a family of good qudit quantum codes supporting transversal logical $\mathsf{C}^{m-1}\mathsf{Z}$ gates that can address specified logical qudits and be largely executed in parallel. Building on the family of good quantum error-correcting codes presented in He et al. (2025), which support addressable and transversal logical $\mathsf{CCZ}$ gates, we extend their framework and show how to perform large sets of gates in parallel. The construction relies on the classical algebraic geometry codes of Stichtenoth (IEEE Trans. Inf. Theory, 2006). Our results lead to a substantial reduction in the depth overhead of multi-control-$Z$ circuits. In particular, we show that the minimal depth of any logical $\mathsf{C}^{m-1}\mathsf{Z}$ circuit involving qudits from $m$ distinct code blocks is upper bounded by $O(k^{m-1})$, where $k$ is the code dimension. While this overhead is optimal for dense $\mathsf{C}^{m-1}\mathsf{Z}$ circuits, for sparse circuits we discuss how the depth overhead can be significantly reduced by exploiting the structure of the quantum code.