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~\cite{silva_clifford_2025}, 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, (\mathbb{Z}_d^2) is a symplectic module rather than a vector space, motivating a distinction between four types of gates. Semi-Clifford gates ($\mathcal{SC}$) are the ones that become diagonal under left and right multiplication by Clifford gates, while a broader class $\mathcal{N}$ admits a permutation diagonal form, under left and right multiplication by Cliffords. Lagrangian semi-Clifford gates ($\mathcal{LSC}$) conjugate some maximal abelian Pauli subgroup to another, whereas generalized semi-Clifford gates ($\mathcal{GSC}$) map some maximal abelian Pauli subalgebra to another under conjugation. In square-free dimensions, we have (\mathcal{SC}=\mathcal{LSC}) and (\mathcal{N}=\mathcal{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.

Source: arXiv:2610.06836v1 - http://arxiv.org/abs/2610.06836v1 PDF: https://arxiv.org/pdf/2610.06836v1 Original Link: http://arxiv.org/abs/2610.06836v1