Quantum simulation of Helmholtz equations via Schr{ö}dingerization

Abstract

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 $\mathcal{O}(\hat{I}{º}^2\text{polylog}{\epsilon }^{-1})$, where $\hat{I}º$ is the condition number and $\epsilon$ the desired accuracy. For the Helmholtz equation, a simple preconditioner further reduces the complexity to $\mathcal{O}(\hat{I}º\text{polylog}{\epsilon }^{-1})$. Our constructive extension to the quantum setting is broadly applicable to all indefinite problems.