We analyze the Schr\"odingerisation method for quantum simulation of a general class of non-unitary dynamics with inhomogeneous source terms. The Schr\"odingerisation technique, introduced in \cite{JLY22a,JLY23}, transforms any linear ordinary and partial differential equations with non-unitary dynamics into a system under unitary dynamics via a warped phase transition that maps the equations into a higher dimension, making them suitable for quantum simulation. This technique can also be applied to these equations with inhomogeneous terms modeling source or forcing terms or boundary and interface conditions, and discrete dynamical systems such as iterative methods in numerical linear algebra, through extra equations in the system. Difficulty airses with the presense of inhomogeneous terms since it can change the stability of the original system. In this paper, we systematically study--both theoretically and numerically--the important issue of recovering the original variables from the Schr\"odingerized equations, even when the evolution operator contains unstable modes. We show that even with unstable modes, one can still construct a stable scheme, yet to recover the original variable one needs to use suitable data in the extended space. We analyze and compare both the discrete and continuous Fourier transforms used in the extended dimension, and derive corresponding error estimates, which allows one to use the more appropriate transform for specific equations. We also provide a smoother initialization for the Schrod\"odingerized system to gain higher order accuracy in the extended space. We homogenize the inhomogeneous terms with a stretch transformation, making it easier to recover the original variable. Our recovering technique also provides a simple and generic framework to solve general ill-posed problems in a computationally stable way.

翻译：我们分析了对一类含非齐次源项的普遍非酉动力学进行量子模拟的薛定谔化方法。薛定谔化技术通过扭曲相位变换将方程映射到更高维空间，从而将非酉动力学的线性常微分方程和偏微分方程转化为酉动力学系统，使其适用于量子模拟。该技术还可通过引入额外方程，应用于带有源项、强迫项、边界条件及界面条件的非齐次方程，以及数值线性代数中的迭代方法等离散动力系统。非齐次项的存在会改变原系统的稳定性，因此带来了处理难度。本文从理论和数值两方面系统研究了从薛定谔化方程中恢复原始变量的关键问题，即便在演化算子含有不稳定模式时依然有效。我们证明，即使存在不稳定模式，仍可构造稳定方案，但需在扩展空间中使用恰当的数据来恢复原始变量。我们分析并比较了在扩展维度上使用的离散傅里叶变换与连续傅里叶变换，推导了相应的误差估计，从而允许针对特定方程选择更合适的变换。我们还为薛定谔化系统提供了更平滑的初始化方法，以在扩展空间中实现更高阶精度。通过伸缩变换对非齐次项进行均匀化处理，简化了原始变量的恢复过程。我们的恢复技术还为以计算稳定方式求解一般病态问题提供了一种简单通用的框架。