Calibrated Pressure-Observable Born and Hessian Actions for Quantum-Assisted Waveform Inversion

We construct a pressure-consistent operator-and-readout interface for Born, adjoint, and Gauss–Newton actions in constant-density acoustic full-waveform inversion (FWI) using Schr"odingerised propagation. The energy variables $\pi =c^{-1}{\mathrm{\partial }}_{t}u$ and $q=\mathrm{\nabla }u$ yield an auxiliary-space Hamiltonian, while physical pressure $p=c\pi$ depends explicitly on wavespeed. Its derivative $D(c\pi )[c_0](\delta c)=c_0\delta \pi +\delta c{\pi }_0$ combines propagated wavefield sensitivity with a direct receiver-calibration term. Duhamel and receiver-row differentiation retain both contributions in the Born map, its adjoint, and the Gauss–Newton normal action. We prove a conditional consistency estimate with a periodic second-order finite-difference specialization and give a resource model for state preparation, normalization, quadrature, and selected-output measurement. A compiled nine-qubit instance realizes structured preparation, product-formula propagation, a derivative-LCU block, and calibrated pressure-overlap measurements. Bernoulli samples from ideal-circuit probabilities drive a four-parameter hybrid inversion. A two-qubit VQLS circuit represents the normalized update direction, while normal-system assembly, line search, and model refresh remain classical. Finite differences, tangent and reverse-adjoint recurrences, autodiff JVP/VJP evaluations, and explicit Jacobians verify the discrete Born, adjoint, and normal actions. Smooth periodic refinement confirms second-order convergence, whereas omitting receiver calibration leaves an order-one Born error and substantially changes the regularized Gauss–Newton direction. All ten predeclared finite-shot runs reduce the initial model error. These results specify the physical-pressure derivative and selected-output measurements needed to connect Schr"odingerised propagation to a local FWI update.