发表机构
École polytechnique fédérale de Lausanne (EPFL); University of California, Berkeley（洛桑联邦理工学院; 加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明单 qudit 的 Clifford 层级中每个门为半-Clifford 当且仅当维度无平方因子，并区分了四种门类型，指出在任意维度下第三级门均为广义半-Clifford。
AI 中文摘要
量子信息理论中的一个重要问题是：Clifford 层级中的每个门是否都是半-Clifford 的，因为这样的门可以通过门隐形传态实现资源高效实现。我们扩展了最近关于素数维度的结果~\cite{silva_clifford_2025}，证明了在维度 \\(d\\) 为无平方因子数时，单个 qudit 的 Clifford 层级中的每个门都是半-Clifford 的，且当且仅当 \\(d\\) 为无平方因子数。在复合维度中，\\(\mathbb{Z}_d^2\\) 是辛模而非向量空间，这促使我们区分四种类型的门。半-Clifford 门（\\(\mathcal{SC}\\)）是指在左乘和右乘 Clifford 门后变为对角的门，而更广泛的类 \\(\mathcal{N}\\) 在左乘和右乘 Clifford 门后允许置换对角形式。拉格朗日半-Clifford 门（\\(\mathcal{LSC}\\)）将某个极大阿贝尔 Pauli 子群共轭到另一个，而广义半-Clifford 门（\\(\mathcal{GSC}\\)）在共轭下将某个极大阿贝尔 Pauli 子代数映射到另一个。在无平方因子维度中，我们有 \\(\mathcal{SC}=\mathcal{LSC}\\) 和 \\(\mathcal{N}=\mathcal{GSC}\\)，但当 \\(d\\) 不是无平方因子数时，这些等价关系可能失效，因为拉格朗日子模可能变为非自由的。我们还证明了在任意维度下，单 qudit Clifford 层级的每个第三级门都是广义半-Clifford 的。
英文摘要
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.
Comments25 pages, 2 figures
