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

arXiv:2610.06836v1 Announce Type: new Abstract: 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.

Abstract: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.

arXiv:2610.06836 [quant-ph]

(or arXiv:2610.06836v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.06836 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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

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