发表机构
Tel Aviv University; Hebrew University of Jerusalem; Weizmann Institute of Science（特拉维夫大学; 耶路撒冷希伯来大学; 魏茨曼科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发展高维无损方法，通过局部到整体定理证明足够不平衡的双侧无损四维立方复形可构造渐近好的量子局部可测试码。
AI 中文摘要
Sipser和Spielman从二分谱展开图或单侧无损展开图构造了LDPC码。在更高维度中，谱展开同样在Dinur、Evra、Livne、Lubotzky和Mozes以及Panteleev和Kalachev构造渐近好的经典LTC和qLDPC码中发挥了核心作用。另外，Lin和Hsieh从二维无损立方复形构造了经典LTC和qLDPC码。在本工作中，我们发展了高维无损方法。我们并不构造所需的高维无损立方复形；而是研究它们的存在会蕴含什么。我们将一个高维立方复形与一个层级链复形相关联，其链群支撑在布尔立方体的水平集上而非其胞腔上。我们的主要技术贡献是针对该结构的一个简洁的局部到整体定理：方向图中的合适的单维无损展开蕴含整体层级复形的小集合同调边界展开。因此，足够不平衡的双侧无损四维立方复形产生渐近好的量子局部可测试码。我们期望这里发展的局部到整体原理有进一步的应用。
英文摘要
Sipser and Spielman constructed LDPC codes from either bipartite \emph{spectral} expanders or one-sided \emph{lossless} expanders. In higher dimensions, \emph{spectral} expansion similarly played a central role in the constructions of asymptotically good classical LTCs and qLDPC codes by Dinur, Evra, Livne, Lubotzky, and Mozes and by Panteleev and Kalachev. Alternatively, Lin and Hsieh constructed classical LTCs and qLDPC codes from two-dimensional \emph{lossless} cubical complexes. In this work we develop the higher-dimensional \emph{lossless} approach. We do not construct the required high-dimensional lossless cubical complexes; rather, we investigate what their existence would imply. We associate with a high-dimensional cubical complex a \emph{level chain complex}, whose chain groups are supported on the level sets of the Boolean cube rather than on its cells. Our main technical contribution is a clean local-to-global theorem for this structure: suitable one-dimensional lossless expansion in the directional graphs implies small-set coboundary expansion of the global level complex. As a consequence, sufficiently imbalanced, two-sided lossless four-dimensional cubical complexes give rise to asymptotically good quantum locally testable codes. We expect the local-to-global principle developed here to have further applications.
Comments44 pages, 5 figures; cleaned up misplaced editorial comments
