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arXiv 2408.12752quant-phcs.ITmath.ITmath.NT
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短长度高距离的横向 Clifford 与 T 门码

Transversal Clifford and T-gate codes of short length and high distance

Shubham P. Jain, Victor V. Albert

备注：
3 tables, 3 figures. Updated version: Added table summarizing T-gate code families and properties. Added discussion on stabilizer weights. Updated title and improved overall presentation
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AI总结：

本文构造了距离最高达31的三类短长度稳定子码，分别支持Clifford群和逻辑T门的横向实现，并在给定距离下实现了已知最小的物理量子比特开销。

摘要：

多种量子设备架构中的非局域相互作用允许在保持相同抗噪声保护程度的同时实现更紧凑的量子编码。考虑到短到中等长度的码不久将可实现，构造在给定码距下能够以最少物理量子比特数实现逻辑门容错实现的稳定子码非常重要。为此，我们构造了三类编码单个逻辑量子比特、距离最高达 $31$ 的码。首先，我们构造了已知最小的 doubly even 码，所有这些码都允许 Clifford 群的横向实现。将倍增程序 [arXiv:1509.03239] 应用于这些码，可以得到相同距离和编码量子比特数下已知最小的 weak triply even 码。这第二类码允许逻辑 $\ exttt{T}$-门的横向实现。放宽 triply even 性质，我们得到第三类 triorthogonal 码，其开销更低，但代价是需要额外的 Clifford 门来实现相同的逻辑操作。据我们所知，这些是各自距离下已知最小的 triorthogonal 码。虽然这些码族不是 qLDPC，但具有横向 $\ exttt{T}$-门的码族的稳定子生成元权重大致按其长度的平方根增长。

英文摘要：

The non-local interactions in several quantum device architectures allow for the realization of more compact quantum encodings while retaining the same degree of protection against noise. Anticipating that short to medium-length codes will soon be realizable, it is important to construct stabilizer codes that, for a given code distance, admit fault-tolerant implementations of logical gates with the fewest number of physical qubits. To this aim, we construct three kinds of codes encoding a single logical qubit for distances up to $31$. First, we construct the smallest known doubly even codes, all of which admit a transversal implementation of the Clifford group. Applying a doubling procedure [arXiv:1509.03239] to such codes yields the smallest known weak triply even codes for the same distances and number of encoded qubits. This second family of codes admit a transversal implementation of the logical $\texttt{T}$-gate. Relaxing the triply even property, we obtain our third family of triorthogonal codes with an even lower overhead at the cost of requiring additional Clifford gates to achieve the same logical operation. To our knowledge, these are the smallest known triorthogonal codes for their respective distances. While not qLDPC, the stabilizer generator weights of the code families with transversal $\texttt{T}$-gates scale roughly as the square root of their lengths.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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