标题： 量子线性系统问题的薛定谔化 显示英文标题

标题： Schrödingerization for quantum linear systems problems

摘要： 我们从Schrödingerization问题的角度开发了一种用于线性代数方程Ax=b的量子算法，这些问题是通过一个更高维度的线性对流方程系统来表征的。 当A是正定矩阵时，解x可以解释为线性ODE的稳态解。 这个ODE可以通过使用[1]中的LCHS方法求解，该方法作为Schrödingerization方法[2,3]中傅里叶变换的连续实现，通过扭曲相位变换将具有非单位动力学的线性PDE和ODE转换为Schrödinger型系统。 与LCHS方法相比，Schrödingerization可能更受PDE社区的欢迎，因为它更适合利用已建立的计算PDE技术来开发量子算法。 当A是一个一般的厄米特矩阵时，逆矩阵仍然可以在[1]中以LCHS形式表示，但其核函数基于[4]中的傅里叶方法。 尽管这种LCHS形式提供了与最小二乘方程相关的线性ODE的稳态解，但将Schrödingerization应用于这种最小二乘法是不合适的，因为它会导致更大的条件数。 我们证明了在两种情况下，解x都可以表示为Schrödingerization问题的LCHS，或者等价地，作为Schrödingerization问题的稳态解。 这突显了Schrödingerization在量子科学计算中的潜力。 我们提供了一个详细的说明，以及几个数值测试来验证我们所提出方法的正确性。 此外，我们开发了一种量子预处理算法，将BPX多级预处理器与我们的方法结合，以解决泊松方程的有限元离散化。 显示英文摘要

摘要： We develop a quantum algorithm for linear algebraic equations Ax=b from the perspective of Schr\"odingerization-form problems, which are characterized by a system of linear convection equations in one higher dimension. When A is positive definite, the solution x can be interpreted as the steady-state solution of linear ODEs. This ODE can be solved by using the LCHS method in [1], which serves as the continuous implementation of the Fourier transform in the Schr\"odingerization method from [2,3] Schr\"odingerization transforms linear PDEs and ODEs with non-unitary dynamics into Schr\"odinger-type systems via the warped phase transformation that maps the equation into one higher dimension. Compared to the LCHS method, Schr\"odingerization may be more appealing to the PDE community, as it is better suited for leveraging established computational PDE techniques to develop quantum algorithms. When A is a general Hermitian matrix, the inverse matrix can still be represented in the LCHS form in [1], but with a kernel function based on the Fourier approach in [4]. Although this LCHS form provides the steady-state solution of linear ODEs associated with the least-squares equation, applying Schr\"odingerization to this least-squares is not appropriate, as it results in a much larger condition number. We demonstrate that in both cases, the solution x can be expressed as the LCHS of Schr\"odingerization-form problems, or equivalently, as the steady-state solution to a Schr\"odingerization-form problem. This highlights the potential of Schr\"odingerization in quantum scientific computation. We provide a detailed, along with several numerical tests that validate the correctness of our proposed method. Furthermore, we develop a quantum preconditioning algorithm that combines the BPX multilevel preconditioner with our method to address the finite element discretization of the Poisson equation.