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

• URL: http://arxiv.org/abs/2505.00370v1
• Date: Thu, 01 May 2025 07:46:50 GMT
• Title: On the Schrödingerization method for linear non-unitary dynamics with optimal dependence on matrix queries
• Authors: Shi Jin, Nana Liu, Chuwen Ma, Yue Yu,
• Abstract summary: 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.
• Score: 30.909666596285067
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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}. We show that by employing a smooth initialization of the warped phase transform \cite{JLM24SchrBackward}, Schr\"odingerization can in fact achieve optimal scaling in matrix queries. This paper presents the detailed implementation of three smooth initializations for the Schr\"odingerization method: (a) the cut-off function, (b) the higher-order polynomial interpolation, and (c) the Fourier transform methods, that achieve optimality for (a) and near-optimality for (b) and (c). A detailed analysis of key parameters affecting time complexity is conducted.

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