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 Schrödingerised propagation.

The energy variables $\pi=c^{-1}\partial_t u$ and $q=\nabla u$ 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 Schrödingerised propagation to a local FWI update.