AI 中文总结
研究量子纠错码族最优纠错阈值，通过统计力学映射证明对偶性约束零速率 em 对称 CSS 码有相同阈值，此自对偶性确定临界点、约束相边界，且在码级联下保持，为分析多种量子码族提供通用框架。
AI 中文摘要
纠错阈值常被视为比较量子纠错码族的主要品质因数。我们表明，在副本极限下，许多常用码的最优纠错阈值在主导阶被约束到一个单一通用值。通过统计力学映射，证明对偶性约束所有零速率的 em 对称 CSS 码具有相同的最优码容量阈值。在此映射下，em 对称 CSS 码在广义 Kramers - Wannier 对偶下是自对偶的。这一自对偶性确定了干净临界点并约束了无序相边界。还表明自对偶性在码级联下保持，级联码的最优解码可重新表述为分层晶格上的重整化群流。我们的结果为分析拓扑、级联及更一般的量子低密度奇偶校验码族提供了通用框架。
英文摘要
Error correction thresholds are often treated as the primary figure of merit for comparing quantum error-correcting code families. We show that the optimal error correction threshold for many commonly considered codes is constrained to a single universal value at leading order in a replica limit. Through a statistical mechanical mapping, we demonstrate that duality constrains all zero-rate em-symmetric CSS codes to have the same optimal code capacity threshold. Here, em symmetry means that the X- and Z-type parity-check matrices are equivalent up to row and column permutations. Under this statistical mechanical mapping, em-symmetric CSS codes are self-dual under a generalized Kramers-Wannier duality up to a mixing of logical sectors. For zero-rate code families, this mixing contributes only subextensive corrections, so the thermodynamic bulk free energy is self-dual in the trivial logical sector. This self-duality fixes the clean critical point and constrains the disordered phase boundary. We also show that self-duality is preserved under code concatenation, and that optimal decoding of concatenated codes can be reformulated as a renormalization group flow on a hierarchical lattice. Our results provide a common framework for analyzing topological, concatenated, and more general quantum low-density parity-check code families, including both their optimal code capacity thresholds and their sub-threshold logical error suppression.
Comments27 pages, comments welcome
