论文标题
横向$ t $门的经典编码问题
Classical Coding Problem from Transversal $T$ Gates
论文作者
论文摘要
通用量子计算需要实现逻辑的非克利福德门。在本文中，我们表征了所有稳定器代码，其代码子空间保存在物理$ t $和$ t^{ - 1} $门下。例如，这可以通过非CSS代码启用魔术状态蒸馏，因此比基于CSS的协议提供了更好的参数。但是，在支持横向$ t $的非分类稳定器代码中，我们证明CSS代码是最佳的。我们还表明，从本质上讲，三串管代码是CSS代码的唯一家族，通过物理横向$ t $实现逻辑横向$ t $。使用我们的代数方法，我们揭示了与通过横向$ t $实现逻辑操作密切相关的新的纯经典编码问题。减少单码也用于构建实现逻辑CCZ的代码。最后，我们使用AX的定理来表征在量子芦苇 - 毛刺代码家族上实现的逻辑操作。该结果概括为ARXIV中的细角$ z $ - 旋转：1910.09333。
Universal quantum computation requires the implementation of a logical non-Clifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical $T$ and $T^{-1}$ gates. For example, this could enable magic state distillation with non-CSS codes and, thus, provide better parameters than CSS-based protocols. However, among non-degenerate stabilizer codes that support transversal $T$, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal $T$ via physical transversal $T$. Using our algebraic approach, we reveal new purely-classical coding problems that are intimately related to the realization of logical operations via transversal $T$. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum Reed-Muller codes. This result is generalized to finer angle $Z$-rotations in arXiv:1910.09333.
