We introduce a high-dimensional cubical complex, for any dimension t>0, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case t=2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes $h_1,\ldots,h_t$ as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. For t=4 our construction gives a new family of "almost-good" quantum LTCs -- with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.

翻译：我们引入一种高维立方复形，适用于任意维度t>0，并将其应用于量子局部可测试码的设计。该复形是Panteleev与Kalachev以及Dinur等人所构造的平方复形（t=2情形）的自然推广，这些构造此前分别被用于经典局部可测试码（LTC）和量子低密度奇偶校验码（qLDPC）的设计。通过采用恒定大小的局部码$h_1,\ldots,h_t$作为构件，我们将几何（立方）复形转化为链复形。Panteleev与Kalachev关于乘积扩张码元组存在性的最新结果，使我们能够证明该链复形中循环与上循环扩张的下界。当t=4时，我们的构造给出了一族新的"准优"量子LTC——具有恒定相对速率、逆多对数级相对距离与可靠性，以及恒定大小的奇偶校验。量子码的距离及其局部可测试性均直接由链复形的循环与上循环扩张推导得出。