Quantum-classical estimation of physically scaled groundwater flow and solute transport from normalized quantum...

Groundwater flow and solute transport models require repeated solutions of large sparse linear systems, for which quantum linear-system algorithms may eventually accelerate computation. However, amplitude encoding produces normalized quantum states rather than physically scaled hydrogeologic variables, creating a readout problem for hydraulic head and concentration fields required for mass- and flux-based analysis. We develop a hybrid quantum-classical estimation framework that combines multipoint constrained rescaling with operator-level transport estimation to recover and propagate physical scale through transient simulations. For well-reversal problems, the framework further incorporates physically prescribed anchors, capture-oriented pumpback observations, and validation-based adaptive regional rescaling. The method is evaluated by classical emulation in a heterogeneous synthetic aquifer and a structured numerical adapter based on the public MODFLOW 6 P06 injection-extraction configuration. In noise-free synthetic tests with exact operators and normalized states, constrained rescaling recovers reference hydraulic head and concentration fields with negligible mass and peak errors, confirming that the dominant missing information is physical scale. In the P06 setting with 64 sampled locations per observation time and 2% observation noise, adaptive regional rescaling reduces concentration RMSE from 0.581 for global multi-point constrained rescaling to 0.0769 under the full stress-period-aware design. With the adaptive method fixed, a budget-matched stress-period-aware design further reduces RMSE from 0.165 under uniform monitoring to 0.0558 under the same observation budget. These results demonstrate the feasibility of converting normalized quantum linear-system outputs into physically scaled groundwater fields suitable for mass- and flux-based hydrogeologic interpretation under controlled benchmark conditions, without establishing robustness to algorithm-specific quantum-solver errors.