CALIBRATED PRESSURE-OBSERVABLE BORN AND HESSIAN ACTIONS FOR QUANTUM-ASSISTED WAVEFORM INVERSION

Full-waveform inversion (FWI) is a cornerstone of seismic imaging and medical ultrasound, but its computational cost is dominated by repeated wave-equation solves. This paper investigates how to implement the necessary Born, adjoint, and Gauss-Newton actions—the building blocks of FWI—using quantum-assisted Schrodingerised propagation. Specifically, the authors address how to correctly define the physical pressure derivative when the wave equation is embedded into a higher-dimensional auxiliary space.

The researchers utilize the energy variables $\pi = c^{-1}\partial_t u$ and $q = \nabla u$ to construct an auxiliary-space Hamiltonian. A critical challenge in this framework is that physical pressure ($p = c\pi$) depends explicitly on the wavespeed $c$. The authors derive the directional derivative of this pressure observable, which consists of two parts: the propagated wavefield sensitivity and a direct receiver-calibration term. They incorporate both terms into the Born map, its adjoint, and the Gauss-Newton normal action. The study validates this construction through a nine-qubit compiled instance, using finite-difference checks, autodiff evaluations, and hybrid quantum-classical inversion runs.

The study proves that the receiver-calibration term is mathematically essential for the consistency of the Born map. Numerical experiments confirm that omitting this term leads to an $O(1)$ error in the Born approximation and fundamentally alters the regularized Gauss-Newton update direction. By including the calibration, the authors achieve second-order convergence in periodic refinement tests. Furthermore, they provide a resource model for state preparation and measurement, demonstrating that their calibrated pressure-observable formulation successfully reduces initial model error in finite-shot hybrid inversion scenarios.