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.

翻译：本文给出了两种具有恒定可靠性的量子局部可测试码（QLTC）构造方案。第一种方法提出了"校验乘积"运算，并展示了该运算如何产生具有恒定可靠性、恒定编码率以及随局域性标度距离的QLTC。第二种方法考虑量子码与经典重复码的超图乘积，观察到分量码可靠性得以保持的特例。该发现引导我们构造出具有恒定可靠性、可扩展编码率与距离、以及恒定平均局域性的QLTC。本工作为构建高可靠性与高距离的QLTC迈出了关键一步，这将为无低能平凡态（NLTS）定理提供另一种构造方案。