论文标题
在Qubit Clifford层次结构中的组上
On Groups in the Qubit Clifford Hierarchy
论文作者
论文摘要
在这里，我们研究了可以使用Qubit Clifford层次结构元素构建的单一群体。我们首先提供了一种必要和足够的规范形式，半甲基和广义的半克利福德要素必须满足在克利福德层次结构中。然后，我们对可以从此类元素形成的组进行分类。直到克利福德共轭，我们对所有这些可以使用Clifford层次结构中的广义半偏度元素进行构建的所有这些组进行了分类。我们讨论了附录中此分类的可能次要例外。这可能不是Qubit Clifford层次结构中所有组的完整分类，因为目前尚不清楚Clifford层次结构中的所有元素是否都必须概括为半决赛。除了Cui等人发现的对角门栅组外，我们还表明，Clifford层次结构中还包含许多非同态（对角栅极组）广义对称组。最后，作为该分类的应用，我们检验了对本文列举的组结构所给出的横向门的限制，这可能是独立的。
Here we study the unitary groups that can be constructed using elements from the qubit Clifford Hierarchy. We first provide a necessary and sufficient canonical form that semi-Clifford and generalized semi-Clifford elements must satisfy to be in the Clifford Hierarchy. Then we classify the groups that can be formed from such elements. Up to Clifford conjugation, we classify all such groups that can be constructed using generalized semi-Clifford elements in the Clifford Hierarchy. We discuss a possible minor exception to this classification in the appendix. This may not be a full classification of all groups in the qubit Clifford Hierarchy as it is not currently known if all elements in the Clifford Hierarchy must be generalized semi-Clifford. In addition to the diagonal gate groups found by Cui et al., we show that many non-isomorphic (to the diagonal gate groups) generalized symmetric groups are also contained in the Clifford Hierarchy. Finally, as an application of this classification, we examine restrictions on transversal gates given by the structure of the groups enumerated herein which may be of independent interest.
