On Schr\"odingerization based quantum algorithms for linear dynamical systems with inhomogeneous terms

TL;DR

This paper studies the Schr"odingerization method for quantum simulation of linear systems with inhomogeneous terms, addressing stability and variable recovery issues, and proposing techniques for improved accuracy and stability.

Contribution

It introduces a systematic approach to recover original variables in Schr"odingerized systems with inhomogeneous terms, including stability analysis and error estimates.

Findings

Stable schemes can be constructed even with unstable modes.

Fourier transform choices affect error and stability.

Smoother initialization improves accuracy.

Abstract

We analyze the Schr\"odingerization method for quantum simulation of a general class of non-unitary dynamics with inhomogeneous source terms. The Schr\"odingerization technique, introduced in [31], 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 arises with the presence of inhomogeneous terms since they 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; however, 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 allow one to use the more appropriate transform for specific equations. We also provide a smoother initialization for the Schr\"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 recovery technique also provides a simple and generic framework to solve general ill-posed problems in a computationally stable way.