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

Quantum Locally Testable Code with Constant Soundness