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：本文分析了一类含非齐次源项的非幺正动力学量子模拟的薛定谔化方法。薛定谔化技术(见文献\cite{JLY22a,JLY23})通过扭曲相变将非幺正动力学的线性常微分和偏微分方程组映射至更高维空间，使其转化为幺正动力学系统，从而适用于量子模拟。该技术还可通过引入附加方程，将含源项、强迫项、边界及界面条件的非齐次项方程，以及数值线性代数中的迭代法等离散动力系统纳入统一处理框架。非齐次项的存在可能改变原系统的稳定性，这构成了主要技术难点。本文从理论和数值两方面系统研究了一个关键问题：即使演化算子包含不稳定模态，如何从薛定谔化方程中恢复原始变量。研究表明，即使存在不稳定模态，仍可构造稳定方案，但需在扩展空间中使用合适的数据来恢复原始变量。我们分析并比较了扩展维度中使用的离散与连续傅里叶变换，推导了相应的误差估计，从而可针对特定方程选择更优变换。此外，我们为薛定谔化系统提出平滑初始化方案，以在扩展空间中获得更高阶精度。通过拉伸变换实现非齐次项均匀化，使原始变量恢复更为简便。本恢复技术还为一般性病态问题的计算稳定求解提供了简洁通用框架。