发表机构
Princeton University; University of Oxford（普林斯顿大学; 牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过扩展共轭群不动点论证，证明了广义半 Clifford 猜想对任意素数维度下 Clifford 层级第 4 层成立，并提出了更高层级门满足该性质的一个充分条件。
AI 中文摘要
Clifford 层级 $\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdots$ 由 Gottesman 和 Chuang (arXiv:quant-ph/9908010) 引入，用于刻画可通过门量子隐形传态实现容错实现的量子门。然而，尽管其数学结构丰富且在近年受到关注，对于 $k > 3$ 的 $\mathsf{C}_k$ 所知甚少。大多数进展集中于识别层级限制的结构性质，例如对角门以及小维度 $d$ 或少量 qudit 系统上的门。广义半 Clifford 猜想由 Zeng 等人 (arXiv:0712.2084) 提出，指出 $\cup_k \mathsf{C}_k$ 中的每个门，在乘以 Clifford 门的意义下，都是一个置换矩阵与一个对角矩阵的乘积。Beigi 和 Shor 证明了 $d=2$, $k=3$ 的情形 (arXiv:0810.5108)，Pllaha 等人通过利用由 $U \in \mathsf{C}_3$（经 Clifford 修正后）在最大稳定子子群张成空间上的共轭映射的不动点，找到了另一种证明方法 (arXiv:2006.14040)。通过将他们的不动点论证扩展到由 $U \mathsf{P} U^\dagger$ 生成的群 $\Gamma_1(U)$ 及其更高层级，我们证明了该猜想对 $k \leq 4$ 和任意素数维度 $d$ 成立。我们的证明核心在于 $U \in \mathsf{C}_k$ 的共轭群 $\Gamma_1(U), \Gamma_2(U), \ldots$，我们预期这将成为研究 Clifford 层级更一般性质的有用工具。我们还展示了这类群的一个自然充分条件，使得更高层级的门成为广义半 Clifford 门。
英文摘要
The Clifford hierarchy $\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdots$ was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about $\mathsf{C}_k$ for $k > 3$. Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension $d$ or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in $\cup_k \mathsf{C}_k$ is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case $d=2$, $k=3$ (arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of) $U \in \mathsf{C}_3$ on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group $Γ_1(U)$ generated by $U \mathsf{P} U^\dagger$ and beyond, we prove the conjecture for $k \leq 4$ and any prime dimension $d$. Our proof centers on conjugation groups $Γ_1(U), Γ_2(U), \ldots$ of $U \in \mathsf{C}_k$, which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.
Comments16 pages, no figures
