Same System, Fewer Measurements: Structured Variational Quantum Linear Solvers

We introduce StructVQLS, a structure-based quantum linear solver exploiting recurring relations in widely used partial differential equation (PDE) discretizations for more accurate solutions with fewer measurements. On a $32768\times 32768$ heat system, GPU circuit-simulator experiments achieve 61.7% lower mean equation residual than merged exact Pauli at 512M measurements per run. Stable 1% residual is reached with StructVQLS, but not with Pauli within this budget in the evaluated cohort. In separate nonsymmetric convection-diffusion experiments, native sampling gives 50.6-60.9% lower mean residual than merged Pauli across two advection weights. Separate measurement emulation achieves 75% fewer measurements with lower final residual than merged Pauli in observed cross-budget comparisons. StructVQLS integrates exact operator completion, residual-preserving distributed coordination and weighted gradient estimation. Fixed stencil relations give grid-independent operator counts; coefficient weights control gradient noise. Common operator structure thus improves measurement efficiency.