论文标题
来自张量的新宇宙膨胀器意味着用$ω（\ sqrt {n} \ log^kn）$ gance明确量子LDPC代码
New Cosystolic Expanders from Tensors Imply Explicit Quantum LDPC Codes with $Ω(\sqrt{n}\log^kn)$ Distance
论文作者
论文摘要
在这项工作中，我们在更高的维度中引入了一种新的扩展概念，该概念比研究良好的宇宙扩张概念更强，并称为{\ em em Collective-CosySystolocy扩展}。 我们表明，假设产品中的一个综合体不仅是宇宙扩张器，而且是一个集体的宇宙延展性扩展器。 然后，我们表明，众所周知的界面宇宙扩张器，Ramanujan综合体实际上是集体宇宙膨胀者。这使我们能够通过张开Ramanujan Complexs来构建新的有限程度的宇宙扩张器。 使用我们的新构造的有限度界体扩展器，我们构建了{\ em explicit}量子LDPC代码$ \ sqrt {n} \ log^k n $的任何$ k $，从而改善了Evra et的最新结果。 al。 \ cite {ekz}，并为显式量子LDPC代码的距离设置新记录。 \ cite {ekz}的工作利用了在Ramanujan Complextes中发生的高维膨胀概念，称为宇宙膨胀。我们的改进是通过考虑Ramanujan Complexs的张量产品，并使用其新衍生的属性（集体循环体育扩展）来实现。
In this work we introduce a new notion of expansion in higher dimensions that is stronger than the well studied cosystolic expansion notion, and is termed {\em Collective-cosystolic expansion}. We show that tensoring two cosystolic expanders yields a new cosystolic expander, assuming one of the complexes in the product, is not only cosystolic expander, but rather a collective cosystolic expander. We then show that the well known bounded degree cosystolic expanders, the Ramanujan complexes are, in fact, collective cosystolic expanders. This enables us to construct new bounded degree cosystolic expanders, by tensoring of Ramanujan complexes. Using our new constructed bounded degree cosystolic expanders we construct {\em explicit} quantum LDPC codes of distance $\sqrt{n} \log^k n$ for any $k$, improving a recent result of Evra et. al. \cite{EKZ}, and setting a new record for distance of explicit quantum LDPC codes. The work of \cite{EKZ} took advantage of the high dimensional expansion notion known as cosystolic expansion, that occurs in Ramanujan complexes. Our improvement is achieved by considering tensor product of Ramanujan complexes, and using their newly derived property, the collective cosystolic expansion.
