Semi-Cliffordness of the Clifford hierarchy for a single qudit in composite dimensions

An important question in quantum information theory asks whether every gate in the Clifford hierarchy is semi-Clifford or not, as such gates admit resource-efficient implementations via gate teleportation. Extending recent results for prime dimensions, we prove that every hierarchy gate on a single qudit of dimension d is semi-Clifford, if and only if d is square-free. In composite dimensions, Z d squared is a symplectic module rather than a vector space, motivating a distinction between four types of gates. Semi-Clifford gates (SC) are the ones that become diagonal under left and right multiplication by Clifford gates, while a broader class N admits a permutation diagonal form, under left and right multiplication by Cliffords. Lagrangian semi-Clifford gates (LSC) conjugate some maximal abelian Pauli subgroup to another, whereas generalized semi-Clifford gates (GSC) map some maximal abelian Pauli subalgebra to another under conjugation. In square-free dimensions, we have SC= LSC and N= GSC, but when d is not square-free, these equivalences can fail, as Lagrangian submodules can become non-free. We also show that every third-level gate of the one-qudit Clifford hierarchy is a generalized semi-Clifford in arbitrary dimensions.