标题： Quantum simulation of Helmholtz equations via Schr{ö}dingerization 显示英文标题

标题： Quantum simulation of Helmholtz equations via Schr{ö}dingerization

摘要： 亥姆霍兹方程是时间谐波传播的典型模型。 数值解在波数$k$增长时变得越来越具有挑战性，这是由于方程的椭圆但非强制特性以及其解的高度振荡性质，波长按$1/k$缩放。 这些特性导致强烈的不定性和大的系统规模。 我们提出了一种用于解决此类不定问题的量子算法，该算法建立在薛定谔化框架之上。 这种方法通过捕捉阻尼动力学的稳态，将线性微分方程重新表述为薛定谔型系统。 一种扭曲相位变换将原始问题提升到更高维的公式，使其与量子计算兼容。 为了抑制数值污染，该算法结合了渐近色散校正。 它实现了查询复杂度为$\mathcal{O}(\kappa^2\text{polylog}\varepsilon^{-1})$，其中$\kappa$是条件数，$\varepsilon$是期望的精度。 对于亥姆霍兹方程，一个简单的预条件器进一步将复杂度降低到$\mathcal{O}(\kappa\text{polylog}\varepsilon^{-1})$。 我们对量子设置的构造性扩展广泛适用于所有不定问题。 显示英文摘要

摘要： 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 Schr\"odingerization framework. This approach reformulates linear differential equations into Schr\"odinger-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 $\mathcal{O}(\kappa^2\text{polylog}\varepsilon^{-1})$, where $\kappa$ is the condition number and $\varepsilon$ the desired accuracy. For the Helmholtz equation, a simple preconditioner further reduces the complexity to $\mathcal{O}(\kappa\text{polylog}\varepsilon^{-1})$. Our constructive extension to the quantum setting is broadly applicable to all indefinite problems.