双变量自行车码距离分析中的逻辑算子分解
Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes
- Centre for Quantum Software and Information, School of Computer Science, University of Technology Sydney（悉尼科技大学计算机科学学院量子软件与信息中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究通过逻辑算子分解，将双变量自行车码的最小距离表示为消去子商与冒号商距离的最小值，并借助 Frobenius 结构及平移锚定簇搜索，证明了六个标准码的距离并枚举了最小权重算子。
AI中文摘要：
双变量自行车（BB）量子码是量子低密度奇偶校验码中一个重要的有限长度族，但其最小距离通常通过数值方法确定，而非从定义多项式直接读出。我们研究 $\mathbb F_2[x,y]/(x^\ell-1,y^m-1)$ 上的 $Z$-逻辑商 $K/S$，并证明它适合一个短正合序列，其中核为消去子商，余核为冒号商。该序列给出了显式逻辑基、维数公式以及分量式距离恒等式 $d_Z=\min(d_{\mathcal A},d_{\mathcal C})$。利用有限群代数的 Frobenius 结构，我们证明对于每个 BB 码（包括重根情形），$r_{\mathcal A}=r_{\mathcal C}=k/2$。逻辑类的代数分量与其最轻代表元的支撑形状不同：消去子类可以有更轻的双块代表元，而冒号类可以具有单侧最小性。对于下界，我们证明最小权重逻辑算子的每个真子集都具有非零校验子，并且该性质在冒号分量内持续成立，但在消去子分量内不成立。基于此构建的平移锚定簇搜索证明了六个标准 BB 码（长度从 18 到 288）的距离分别为 $4,6,10,10,12,18$，并枚举了所有最小权重逻辑算子。由此得到的普查表明，$[[108,8,10]]$ 是六个码中唯一一个距离在单一分量中达到的码，其中 $d_{\mathcal C}=10$，$d_{\mathcal A}=12$。
英文摘要：
Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the $Z$-logical quotient $K/S$ over $\mathbb F_2[x,y]/(x^\ell-1,y^m-1)$ and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity $d_Z=\min(d_{\mathcal A},d_{\mathcal C})$. Using the Frobenius structure of the finite group algebra, we prove $r_{\mathcal A}=r_{\mathcal C}=k/2$ for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with $d_{\mathcal C}=10$ and $d_{\mathcal A}=12$.
