提升乘积码
Lifting Lifted Product Codes
AI总结：
本文提出基于群扩张和图提升的LP码族系统构造，获更优LP码参数，可转移码手术小工具且开销更低，开发并行乘积手术，为无欧几里得晶格的qLDPC码定义热力学族，相干信息现有限尺寸交叉。
AI中文摘要：
提升乘积（LP）码是一类具有优良码参数的重要量子纠错码。我们引入了基于群扩张和图提升的LP码族的系统构造方法，该构造在增大码尺寸的同时保留了Tanner图的局部结构，并通过链映射和上链映射关联了码族内的码参数、逻辑算子及容错逻辑操作小工具。作为首个应用，我们得到了比此前报道的参数更优的LP码；随后证明，码手术小工具可通过链映射在选定的有限提升间转移，且在多种情形下能以更低的空间开销实现；我们还开发了用于提升的簇循环码的并行乘积手术。最后，我们提出将提升作为系统的第一步，以定义无基础欧几里得晶格的代数定义qLDPC码的热力学族。对于若干基码和选定的提升，相干信息呈现有限尺寸交叉，我们的结果还表明需要额外条件来确定唯一的码族。
英文摘要：
Lifted product (LP) codes form an important class of quantum error correcting codes with favorable code parameters. We introduce a systematic construction of LP code families based on group extensions and graph lifts. The construction increases the code size while preserving the local structure of the Tanner graph, and relates code parameters, logical operators, and fault-tolerant logical-operation gadgets within the families through chain and cochain maps. As a first application, we obtain LP codes with better code parameters than previously reported ones. We then demonstrate that code-surgery gadgets can be transferred across the selected finite lifts through chain maps and, in several cases, implemented with lower space overhead. We also develop parallel product surgery for lifted clustered cyclic codes. Finally, we propose lifting as a systematic first step toward defining thermodynamic families for algebraically defined qLDPC codes without an underlying Euclidean lattice. For several base codes and selected lifts, coherent information exhibits finite-size crossings, while our results also indicate that additional conditions are needed to determine a unique family.
