We examine the following problem: given a collection of Clifford gates, describe the set of unitaries generated by circuits composed of those gates. Specifically, we allow the standard circuit operations of composition and tensor product, as well as ancillary workspace qubits as long as they start and end in states uncorrelated with the input, which rule out common "magic state injection" techniques that make Clifford circuits universal. We show that there are exactly 57 classes of Clifford unitaries and present a full classification characterizing the gate sets which generate them. This is the first attempt at a quantum extension of the classification of reversible classical gates introduced by Aaronson et al., another part of an ambitious program to classify all quantum gate sets.
The classification uses, at its center, a reinterpretation of the tableau representation of Clifford gates to give circuit decompositions, from which elementary generators can easily be extracted. The 57 different classes are generated in this way, 30 of which arise from the single-qubit subgroups of the Clifford group. At a high level, the remaining classes are arranged according to the bases they preserve. For instance, the CNOT gate preserves the X and Z bases because it maps X-basis elements to X-basis elements and Z-basis elements to Z-basis elements. The remaining classes are characterized by more subtle tableau invariants; for instance, the T_4 and phase gate generate a proper subclass of Z-preserving gates.

中文翻译：

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克利福德盖茨对量子比特的分类

我们研究以下问题：给定一组 Clifford 门，描述由这些门组成的电路生成的酉集。具体来说，我们允许组合和张量积的标准电路操作，以及辅助工作区量子位，只要它们以与输入不相关的状态开始和结束，这排除了使克利福德电路通用的常见“魔法状态注入”技术。我们展示了恰好有 57 个 Clifford 酉类，并提供了一个完整的分类来表征生成它们的门集。这是 Aaronson 等人引入的可逆经典门分类的第一次量子扩展尝试，这是对所有量子门集进行分类的雄心勃勃计划的另一部分。
该分类在其中心使用对克利福德门的表格表示的重新解释，以给出电路分解，从中可以轻松提取基本生成器。以这种方式生成了 57 个不同的类，其中 30 个来自 Clifford 群的单量子比特子群。在高层次上，剩余的类是根据它们保留的基础进行排列的。例如，CNOT 门保留 X 和 Z 基，因为它将 X 基元素映射到 X 基元素，将 Z 基元素映射到 Z 基元素。其余类的特点是更微妙的画面不变量；例如，T_4 和相位门生成一个适当的 Z 保留门子类。