Semi-Cliffordness of the Clifford hierarchy for a single qudit in composite 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, ℤd2 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.
Related
- When is the Clifford hierarchy generalized semi-Clifford?
- A Cohomological Characterization of the Clifford Hierarchy
- Clifford-hierarchy stabilizer formalism with applications to twisted quantum doubles
- A Complete and Natural Rule Set for Multi-Qudit Clifford Circuits in All Odd Prime Dimensions
- Three-Body Holonomy as a Toy-Model Mechanism for Family Triplication and Mass Hierarchy in (1+1) Dimensions
BibTeX
@misc{qi2026semicliffordness,
title = {{Semi-Cliffordness of the Clifford hierarchy for a single qudit in composite dimensions}},
author = {Yifei Qi and Rahul Sarkar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06836},
eprint = {2610.06836},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06836},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Semi-Cliffordness of the Clifford hierarchy for a single qudit in composite dimensions
AU - Yifei Qi
AU - Rahul Sarkar
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06836
UR - https://arxiv.org/abs/2610.06836
AB - 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, ℤ_d^2 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.
ER -