Schrödingerization for quantum linear systems problems with near-optimal dependence on matrix queries

[2508.13510] Schrödingerization for quantum linear systems problems with near-optimal dependence on matrix queries Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Quantum Physics arXiv:2508.13510 (quant-ph) [Submitted on 19 Aug 2025 ( v1 ), last revised 11 Apr 2026 (this version, v2)] Title: Schrödingerization for quantum linear systems problems with near-optimal dependence on matrix queries Authors: Yin Yang , Yue Yu , Long Zhang View a PDF of the paper titled Schr\"odingerization for quantum linear systems problems with near-optimal dependence on matrix queries, by Yin Yang and Yue Yu and Long Zhang View PDF HTML (experimental) Abstract: We develop a quantum algorithm for linear algebraic equations $ A\bb{x} = \bb{b} $ from the perspective of Schrödingerization-form problems, which are characterized by a system of linear convection equations in one higher dimension. When $ A $ is positive definite, the solution $ \bb{x} $ can be interpreted as the steady-state solution to a system of linear ordinary differential equations (ODEs). This ODE system can be solved by using the linear combination of Hamiltonian simulation (LCHS) method in \cite{ACL2023LCH2}, which serves as the continuous implementation of the Fourier transform in the Schrödingerization method from \cite{JLY22SchrShort, JLY22SchrLong}. Schrödingerization transforms linear partial differential equations (PDEs) and ODEs with non-unitary dynamics into Schrödinger-type systems via the so-called warped phase transformation that maps the equation into one higher dimension. When $ A $ is a general Hermitian matrix, the inverse matrix can still be represented in the LCHS form in \cite{ACL2023LCH2}, but with a kernel function based on the Fourier approach in \cite{Childs2017QLSA}. Although this LCHS form provides the steady-state solution to a system of linear ODEs associated with the least-squares equation, applying Schrödingerization to this least-squares system is not appropriate, as it results in a much larger condition number. We demonstrate that in both cases, the solution $ \bb{x} $ can be expressed as the LCHS of Schrödingerization-form problems. We provide a detailed implementation and error analysis. Furthermore, we incorporate a block preconditioning technique to achieve nearly linear scaling in the condition number, thereby attaining near-optimal query complexity. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2508.13510 [quant-ph] (or arXiv:2508.13510v2 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2508.13510 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Long Zhang [ view email ] [v1] Tue, 19 Aug 2025 04:48:45 UTC (721 KB) [v2] Sat, 11 Apr 2026 08:19:49 UTC (29 KB) Full-text links: Access Paper: View a PDF of the paper titled Schr\"odingerization for quantum linear systems problems with near-optimal dependence on matrix queries, by Yin Yang and Yue Yu and Long Zhang View PDF HTML (experimental) TeX Source view license Current browse context: quant-ph < prev | next > new | recent | 2025-08 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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