The Schr\"odingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schr\"odinger-type equations with unitary evolution. It does so via the so-called warped phase transformation that maps the original equation into a Schr\"odinger-type equation in one higher dimension \cite{Schrshort,JLY22SchrLong}. The original proposal used a particular initial function in the auxiliary space that did not achieve optimal scaling in precision. Here we show that, by choosing smoother initial functions in auxiliary space, Schr\"odingerization \textit{can} in fact achieve near optimal and even optimal scaling in matrix queries. We construct three necessary criteria that the initial auxiliary state must satisfy to achieve optimality. This paper presents detailed implementation of four smooth initializations for the Schr\"odingerization method: (a) the error function and related functions, (b) the cut-off function, (c) the higher-order polynomial interpolation, and (d) Fourier transform methods. Method (a) achieves optimality and methods (b), (c) and (d) can achieve near-optimality. A detailed analysis of key parameters affecting time complexity is conducted.

翻译：薛定谔化方法通过所谓的扭曲相变换，将具有非幺正动力学的线性偏微分方程和常微分方程转化为具有幺正演化的薛定谔型方程组。该方法将原始方程映射到高一维空间中的薛定谔型方程 \cite{Schrshort,JLY22SchrLong}。原始方案在辅助空间中使用了特定的初始函数，未能实现精度上的最优标度。本文证明，通过在辅助空间中选择更光滑的初始函数，薛定谔化方法实际上能够实现接近最优甚至最优的矩阵查询标度。我们构建了辅助初始态为达到最优性必须满足的三个必要准则。本文详细阐述了薛定谔化方法的四种光滑初始化实现方案：(a) 误差函数及相关函数，(b) 截断函数，(c) 高阶多项式插值，以及(d) 傅里叶变换方法。方法(a)达到最优性，方法(b)、(c)和(d)可达到接近最优性。文中对影响时间复杂度的关键参数进行了详细分析。