We introduce a higher-dimensional "cubical" chain complex and apply it to the design of quantum locally testable codes. Our cubical chain complex can be constructed for any dimension $t$, and in a precise sense generalizes the Sipser-Spielman construction of expander codes (case $t=1$) and the constructions by Dinur et. al and Panteleev and Kalachev of a square complex (case $t$=2), which have been applied to the design of classical locally testable and quantum low-density parity check codes respectively. For $t=4$ our construction gives a family of quantum locally testable codes conditional on a conjecture about robustness of four-tuples of random linear maps. These codes have linear dimension, inverse poly-logarithmic relative distance and soundness, and polylogarithmic-size parity checks. Our complex can be built in a modular way from two ingredients. Firstly, the geometry (edges, faces, cubes, etc.) is provided by a set $G$ of size $N$, together with pairwise commuting sets of actions $A_1,\ldots,A_t$ on it. Secondly, the chain complex itself is obtained by associating local coefficient spaces based on codes, with each geometric object, and introducing local maps on those coefficient spaces. We bound the cycle and co-cycle expansion of the chain complex. The assumptions we need are two-fold: firstly, each Cayley graph $Cay(G,A_j)$ needs to be a good (spectral) expander, and secondly, the families of codes and their duals both need to satisfy a form of robustness (that generalizes the condition of agreement testability for pairs of codes). While the first assumption is easy to satisfy, it is currently not known if the second can be achieved.

翻译：我们引入了一种高维“立方”链复形，并将其应用于量子局部可测试码的设计。我们的立方链复形可针对任意维度$t$构造，并在精确意义上推广了扩展器码的Sipser-Spielman构造（情形$t=1$）以及Dinur等人和Panteleev与Kalachev的方形复形构造（情形$t=2$），这些构造分别应用于经典局部可测试码和量子低密度奇偶校验码的设计。对于$t=4$，我们的构造在关于随机线性映射四元组鲁棒性的一个猜想下，得到一族量子局部可测试码。这些码具有线性维度、逆多对数相对距离与稳健性，以及多对数大小的奇偶校验。我们的复形可通过两个组成部分以模块化方式构建。首先，几何结构（边、面、立方体等）由大小为$N$的集合$G$及其上的两两交换作用集$A_1,\ldots,A_t$提供。其次，链复形本身通过将基于码的局部系数空间关联到每个几何对象，并引入这些系数空间上的局部映射而得到。我们给出了该链复形的环与上环扩张的界。所需假设分为两层：首先，每个Cayley图$Cay(G,A_j)$需为良好的（谱）扩展器；其次，码族及其对偶码族均需满足某种鲁棒性形式（该形式推广了码对的协议可测试性条件）。尽管第一项假设易于满足，但第二项假设目前尚不可实现。