论文标题
量子制革商代码
Quantum Tanner codes
论文作者
论文摘要
坦纳代码是从短代码和图形获得的长期错误纠正代码，边缘和奇偶校验检查的限制是从图形顶点强制执行的短代码。将良好的简短代码与光谱扩展器组合在一起，产生了著名的Sipser和Spielman的扩展器代码，这些代码在渐近的古典LDPC代码上是渐进的。 在这项工作中，我们将此处方应用于左右Cayley综合体，这是Dinur等人最近建造的$ C^3 $本地测试代码的核心。具体来说，我们将此复合物视为共享相同边缘的两个图。通过在每个图表上定义坦纳代码，我们获得了两个经典代码，它们共同定义了量子代码。该结构可以看作是Panteleev和Kalachev渐近质量LDPC代码的简化变体，对其最小距离进行了改进的估计。该量子代码与Dinur等人密切相关。从多个意义上讲，代码：实际上，我们证明了一个定理，该定理同时给出了量子代码的线性最小距离，并恢复了Dinur等人的局部测试性。代码。
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.
