On Schrödingerization based quantum algorithms for linear dynamical systems with inhomogeneous terms

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.