The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and ‘nearly diagonal’ semi-Clifford gates are particularly important: they admit efficient gate teleportation protocols that implement these gates with fewer ancillary quantum resources such as magic states. Despite the practical importance of these sets of gates, many questions about their structure remain open; this is especially true in the higher-dimensional qudit setting. Our contribution is to leverage the discrete Stone–von Neumann theorem and the symplectic formalism of qudit stabilizer theory towards extending the results of Zeng et al. (2008) and Beigi & Shor (2010) to higher dimensions in a uniform manner. We further give a simple algorithm for recursively enumerating all gates of the Clifford hierarchy, a simple algorithm for recognizing and diagonalizing semi-Clifford gates, and a concise proof of the classification of the diagonal Clifford hierarchy gates due to Cui et al. (2016) for the single-qudit case. We generalize the efficient gate teleportation protocols of semi-Clifford gates to the qudit setting and prove that every third-level gate of one qudit (of any prime dimension) and of two qutrits can be implemented efficiently. Numerical evidence gathered via the aforementioned algorithms supports the conjecture that higher-level gates can be implemented efficiently.

Clifford 层次结构是一组嵌套的量子门序列，对于实现容错量子计算至关重要。Clifford 层次结构的对角线门和“接近对角线”的半 Clifford 门特别重要：它们允许使用有效的门隐形传输协议，以较少的辅助量子资源（例如魔术状态）来实现这些门。尽管这些门组具有实际重要性，但关于其结构的许多问题仍然悬而未决。在高维量子比特设置中尤其如此。我们的贡献是利用离散的斯通-冯诺依曼定理和量子稳定器理论的辛形式主义来扩展 Zeng等人的结果. (2008) 和 Beigi & Shor (2010) 以统一的方式扩展到更高的维度。我们进一步给出了一个递归枚举 Clifford 层级所有门的简单算法，一个识别和对角化半 Clifford 门的简单算法，以及 Cui等人对对角 Clifford 层级门的分类的简明证明。(2016) 用于单量子比特案例。我们将半克利福德门的有效门传送协议推广到 qudit 设置，并证明一个 qudit（任何素数维度）和两个 qutrit 的每个三级门都可以有效地实现。通过上述算法收集的数字证据支持可以有效实现更高级别门的猜想。