On the Schrödingerization method for linear non-unitary dynamics with optimal dependence on matrix queries
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917128567783424 |
|---|---|
| author | Jin, Shi Liu, Nana Ma, Chuwen Peng, Yizhe Yu, Yue |
| author_facet | Jin, Shi Liu, Nana Ma, Chuwen Peng, Yizhe Yu, Yue |
| contents | The Schrödingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrödinger-type equations with unitary evolution. It does so via the so-called warped phase transformation that maps the original equation into a Schrödinger-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, Schrödingerization \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 Schrödingerization 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00370 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Schrödingerization method for linear non-unitary dynamics with optimal dependence on matrix queries Jin, Shi Liu, Nana Ma, Chuwen Peng, Yizhe Yu, Yue Numerical Analysis Quantum Physics The Schrödingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrödinger-type equations with unitary evolution. It does so via the so-called warped phase transformation that maps the original equation into a Schrödinger-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, Schrödingerization \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 Schrödingerization 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. |
| title | On the Schrödingerization method for linear non-unitary dynamics with optimal dependence on matrix queries |
| topic | Numerical Analysis Quantum Physics |
| url | https://arxiv.org/abs/2505.00370 |