Efficient quantum gate teleportation in higher dimensions
Efficient quantum gate teleportation in higher dimensions
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DOI：
10.1098/rspa.2020.0865
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发表时间：
2020-10
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影响因子：
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通讯作者：
Nadish de Silva
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作者：
Nadish de Silva
Clifford层次结构是一组量子门的嵌套序列，对于实现容错量子计算至关重要。Clifford层次的对角门和“近对角”半Clifford门是特别重要的：他们承认有效的门隐形传态协议，实现这些门与更少的辅助量子资源，如魔术状态。尽管这些门的实际重要性，许多关于它们的结构的问题仍然是开放的;这在更高维度的量子点设置中尤其如此。我们的贡献是利用离散的Stone-von Neumann定理和量子稳定器理论的辛形式主义，以统一的方式将Zeng et al.（2008）和Beigi & Shor（2010）的结果扩展到更高的维度。我们进一步给出了一个递归枚举Clifford层次所有门的简单算法，一个识别和对角化半Clifford门的简单算法，以及由于Cui et al.（2016）对于单量子点情况对角Clifford层次门的分类的简明证明。我们将半Clifford门的有效门传送协议推广到qudit设置，并证明了一个qudit（任何素数维）和两个qutrits的每一个第三级门都可以有效地实现。通过上述算法收集的数值证据支持更高级别的门可以有效地实现的猜想。
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.