In this paper, we present two constructions of quantum locally testable codes (QLTC) with constant soundness. In the first approach, we introduce an operation called check product, and show how this operation gives rise to QLTCs of constant soundness, constant rate, and distance scaling with locality. In the second approach, we consider hypergraph product of a quantum code and a classical repetition code, and observe a special case in which the soundness of component codes is preserved. This insight leads us to construct QLTCs of constant soundness, scalable rate and distance, and constant average locality. Our work marks a step towards constructing QLTCs of high soundness and distance, which would give a different construction to the No Low-Energy Trivial States (NLTS) theorem.

中文翻译：

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具有恒定健全性的 Quantum 本地可测试代码


在本文中，我们提出了两种具有恒定健全性的量子局部可测试代码 （QLTC） 结构。在第一种方法中，我们引入了一个名为 check product 的操作，并展示了该操作如何产生具有恒定健全性、恒定速率和随局部性进行距离缩放的 QLTC。在第二种方法中，我们考虑量子代码和经典重复代码的超图乘积，并观察到一种特殊情况，其中组件代码的健全性被保留。这一见解引导我们构建具有恒定稳健性、可扩展速率和距离以及恒定平均局部性的 QLTC。我们的工作标志着朝着构建高可靠性和距离的 QLTC 迈出了一步，这将为无低能量琐碎态 （NLTS） 定理提供不同的构造。