AI 中文总结
该研究确定了量子译码中MP与MLD译码逻辑获胜集分离的物理错误权重，定义了熵刚性深度r并构建三级层级，为低噪声译码器选择提供精确基准。
AI 中文摘要
最大概率（MP）译码选择最可能的微观错误，而简并最大似然（MLD）译码则包含整个逻辑扇区的构型熵。采用码容量泡利噪声来分离代码固有的刚性，我们确定了其逻辑获胜集变得不相交的首个物理错误权重m，即使在最优MP平局解析下亦是如此。码距施加了通用界m≥h=⌈d/2⌉。我们通过m=h+r定义熵刚性深度r，并验证了一个三级层级：平面表面码及两个级联族的r=0；奇距平方环面码与Gross [[144,12,12]]量子低密度奇偶校验码的r=1；超图积与双变量自行车描述的可分集的r=2。该 onset 确定了主导的操作失效间隙，其与物理噪声强度的m次幂成正比。因此，几何与代数为构型熵提供了可量化的控制，并为低噪声译码器选择提供了精确基准。
英文摘要
Maximum-probability (MP) decoding selects the most probable microscopic error, whereas degenerate maximum-likelihood (MLD) decoding includes the configurational entropy of an entire logical sector. Using code-capacity Pauli noise to isolate rigidity intrinsic to the code, we determine the first physical error weight $m$ at which their logical winner sets become disjoint, even under optimal MP tie resolution. Code distance imposes the universal bound $m\geq h=\lceil d/2\rceil$. We define the entropic rigidity depth $r$ through $m=h+r$ and certify a three-level hierarchy: $r=0$ for planar surface codes and two concatenated families, $r=1$ for odd-distance square toric codes and the Gross $[[144,12,12]]$ quantum low-density-parity-check code, and $r=2$ for a separable family with hypergraph product and bivariate bicycle descriptions. The onset fixes the leading operational failure gap, proportional to the $m$th power of the physical noise strength. Geometry and algebra therefore provide quantifiable controls of configurational entropy and an exact benchmark for low-noise decoder selection.
Comments36 pages, 6 figures. Supplemental Material included
