On the Schrödingerization method for linear non-unitary dynamics with optimal dependence on matrix queries

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.