AI 中文总结
该研究将提升手术扩展到非阿贝尔双块群代数码，证明非交换性带来的测量增益至多为最大阿贝尔子群指数，并找到增益为2和3的码，在电路模拟中非阿贝尔装置更可靠且轮数更少。
AI 中文摘要
代码手术通过$O(d)$轮综合征提取来测量量子LDPC码的逻辑算子。提升手术通过在合并码中沿平移对称性测量逻辑算子的轨道，摊销了阿贝尔群代数码上的这一成本。我们将其扩展到非阿贝尔群上的双块群代数码，并探究非交换性是否能让一个合并码比其对称性的任何阿贝尔群测量更多算子。我们分类了这些码的块仿射自同构，并发现，相对于这个完整群来衡量，大多数表面上的非阿贝尔增益消失了。我们证明增益至多为最大阿贝尔子群的指数。我们找到了精确距离至多13的码，其中增益为2，包括一个合并码读出每个逻辑量子比特的码，以及基于$A_4$、$S_4$和SL(2,3)乘积的码，其最佳逻辑增益为3，达到$[[336,26,12]]$。一个合并距离引理给出了该装置保持码距离的简单条件。在采用Relay-BP解码的电路级模拟中，在我们能分辨的错误率下，非阿贝尔装置与它所替代的阿贝尔装置在统计误差内同样可靠或更可靠，同时使用的轮数少两到三倍。
英文摘要
Code surgery measures a logical operator of a quantum LDPC code with $O(d)$ rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to $13$ where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of $A_4$, $S_4$ and SL(2,3) with gain three for a best logical, up to $[[336,26,12]]$. A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.
Comments25 pages, 3 figures, 8 tables
