arXiv 2202.136412022-02quant-phcs.CCcs.ITmath.IT

量子泰纳码

Quantum Tanner codes

Anthony Leverrier, Gilles Zémor

备注：: 35 pages, v2: added a description of the quadripartite version of the code and an almost matching upper bound for the distance, v3: full version of the extended abstract appearing at FOCS 2022

摘要：

Tanner码是从短码和图中获得的长纠错码，其中位在边缘上，并且来自短码的奇偶校验约束在图的顶点处实施。将好的短码与谱扩展图相结合，得到了著名的扩展码Sipser和Spielman，它们是渐近好的经典LDPC码。在这项工作中，我们将此处方的左，右凯莱复杂的心脏在于最近建设的一个$c^3$本地可测试的代码Dinur等人。具体来说，我们认为这个复杂的两个图形，共享相同的一组边缘。通过在每个图上定义一个Tanner码，我们得到了两个经典码，它们一起定义了一个量子码。这种构造可以被看作是Panteleev和Kalachev渐近好的量子LDPC码的简化变体，其最小距离的估计得到了改进。这种量子码是密切相关的Dinur等人。代码在不止一个意义上：事实上，我们证明了一个定理，同时给出了一个线性增长的最小距离的量子码和恢复的本地可测性Dinur等人。代码。

英文摘要：

Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes. In this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a $c^3$ locally testable code by Dinur et al. Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur et al. code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et al. code.

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。

摘要：

英文摘要：

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