Relations Among Listed Pauli Terms Bound Distance in Qubit Stabilizer qLTCs

Abstract

How many linear relations must a quantum locally testable stabilizer Hamiltonian contain? We prove a finite relation–distance law for qubit stabilizer presentations that bounds code distance in terms of local-term weight, code rate, normalized soundness, and the dimension of the relation space among listed terms. Thus, under bounded term weight, a family cannot simultaneously have constant rate, linear distance, and constant normalized soundness without linearly many independent relations. With bounded qubit incidence, this remains true after duplicate terms are removed. For Calderbank–Shor–Steane presentations with constant normalized soundness in both check sectors, the requirement separates by sector, so redundancy cannot be concentrated in one sector. The result provides a rank- computable necessary test for candidate locally testable stabilizer Hamiltonians, a soundness ceiling for natural rank-minimal hypergraph-product presentations, and sectorwise lower bounds for term-appending amplifiers.