稳定子码的MacWilliams扩展定理的极小反例
Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes
浏览论文内容
中文总结 AI 辅助
该研究针对稳定子码的MacWilliams扩展定理，在尽可能小的尺度上构造了反例，证明其不成立，并确定了不同类型反例的极小长度，还提出了相关开放问题。
中文摘要 AI 辅助
MacWilliams扩展定理对于具有非单根基的模字母表不成立，而量子位稳定子码的标签字母表——有限域F_q上的F_{q²}就是这样的字母表。然而，量子纠错仅涉及自正交加法码，这种刚性是否能挽救该定理，等价于：稳定子群的每个保权同构是否由局部Clifford算子和量子位置换实现，这一问题由Gluesing-Luerssen和Pllaha提出，Pllaha针对特定量子位码给出了否定回答。我们系统地且在尽可能小的尺度上推导了这一否定结论：对每个素数幂q，我们构造了一对[[q+1,q-1]]_q稳定子码，以及它们之间的一个保权同构，该同构不能扩展为单项式变换；尽管这些码空间具有相同的Shor–Laflamme计数，但在任意局部幺正算子与置换的组合下仍不等价。此处的自正交性由两个基本引理自动保证，这两个引理还表明Dyshko的阈值长度反例已具备自正交性，只是未被注意到。对于量子位，我们通过穷举搜索证明长度3是极小的，且该反例本质上是唯一的。我们舍弃了检测这些反例的“空闲量子位”不变量，得到了极小的全支撑长度：非可扩展等距的极小长度为4，非单项式等价的保权等距对的极小长度为5，由显式[[5,2]]码实现；长度为6时，所有非平凡稳定子元素的重量都可≥4。这些码空间是否局部幺正等价是一个开放问题，它将扩展问题与LU–LC相关问题联系起来。
英文摘要
The MacWilliams extension theorem fails for module alphabets with non-cyclic socle, and the label alphabet of qudit stabilizer codes, $\F_{q^2}$ over $\F_q$, is such an alphabet. Quantum error correction, however, only ever sees \emph{self-orthogonal} additive codes, and whether that rigidity rescues the theorem---equivalently, whether every weight-preserving isomorphism of stabilizer groups is implemented by local Cliffords and a qudit permutation---was asked by Gluesing-Luerssen and Pllaha and answered negatively by Pllaha for particular qubit codes. We develop the negative answer systematically and at the smallest possible scales. For every prime power $q$ we construct a pair of $[[q+1,q-1]]_q$ stabilizer codes and a weight-preserving isomorphism between them extending to no monomial transformation; the codespaces are inequivalent even under arbitrary local unitaries combined with permutations, though they share Shor--Laflamme enumerators. Self-orthogonality is automatic here, by two elementary lemmas which also show that Dyshko's threshold-length counterexamples were already self-orthogonal, unremarked. For qubits we prove by exhaustive search that length $3$ is minimal and the counterexample essentially unique. Dropping the ``idle qudit'' invariant that detects these, we find the minimal full-support lengths: $4$ for a non-extendable isometry, $5$ for a weight-isometric pair that is not monomially equivalent, realized by explicit $[[5,2]]$ codes; at length $6$ all nontrivial stabilizer elements can have weight $\ge 4$. Whether these codespaces are locally unitarily equivalent is posed as an open problem, connecting the extension problem to the LU--LC circle of questions.
