Quantum simulation of Helmholtz equations via Schr{ö}dingerization

The Helmholtz equation is a prototypical model for time-harmonic wave propagation. Numerical solutions become increasingly challenging as the wave number k grows, due to the equation's elliptic yet noncoercive character and the highly oscillatory nature of its solutions, with wavelengths scaling as $1/k$. These features lead to strong indefiniteness and large system sizes. We present a quantum algorithm for solving such indefinite problems, built upon the Schrödingerization framework. This approach reformulates linear differential equations into Schrödinger-type systems by capturing the steady state of damped dynamics. A warped phase transformation lifts the original problem to a higher-dimensional formulation, making it compatible with quantum computation. To suppress numerical pollution, the algorithm incorporates asymptotic dispersion correction. It achieves a query complexity of O(κ²polylogε⁻¹), where $κ$ is the condition number and ε the desired accuracy. For the Helmholtz equation, a simple preconditioner further reduces the complexity to O(κpolylogε⁻¹). Our constructive extension to the quantum setting is broadly applicable to all indefinite problems.