Quantum preconditioning method for linear systems problems via Schrödingerization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912370015600640 |
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| author | Jin, Shi Liu, Nana Ma, Chuwen Yu, Yue |
| author_facet | Jin, Shi Liu, Nana Ma, Chuwen Yu, Yue |
| contents | We present a quantum computational framework that systematically converts classical linear iterative algorithms with fixed iteration operators into their quantum counterparts using the Schrödingerization technique [Shi Jin, Nana Liu and Yue Yu, Phys. Rev. Lett., vol. 133 No. 230602,2024]. This is achieved by capturing the steady state of the associated differential equations. The Schrödingerization technique transforms linear partial and ordinary differential equations into Schrödinger-type systems, making them suitable for quantum computing. This is accomplished through the so-called warped phase transformation, which maps the equation into a higher-dimensional space. Building on this framework, we develop a quantum preconditioning algorithm that leverages the well-known BPX multilevel preconditioner for the finite element discretization of the Poisson equation. The algorithm achieves a near-optimal dependence on the number of queries to our established input models, with a complexity of $\mathscr{O}(\text{polylog} \frac{1}{\varepsilon})$ for a target accuracy of $\varepsilon$ when the dimension $d\geq 2$. This improvement results from the Hamiltonian simulation strategy applied to the Schrödingerized preconditioning dynamics, coupled with the smoothing of initial data in the extended space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum preconditioning method for linear systems problems via Schrödingerization Jin, Shi Liu, Nana Ma, Chuwen Yu, Yue Numerical Analysis We present a quantum computational framework that systematically converts classical linear iterative algorithms with fixed iteration operators into their quantum counterparts using the Schrödingerization technique [Shi Jin, Nana Liu and Yue Yu, Phys. Rev. Lett., vol. 133 No. 230602,2024]. This is achieved by capturing the steady state of the associated differential equations. The Schrödingerization technique transforms linear partial and ordinary differential equations into Schrödinger-type systems, making them suitable for quantum computing. This is accomplished through the so-called warped phase transformation, which maps the equation into a higher-dimensional space. Building on this framework, we develop a quantum preconditioning algorithm that leverages the well-known BPX multilevel preconditioner for the finite element discretization of the Poisson equation. The algorithm achieves a near-optimal dependence on the number of queries to our established input models, with a complexity of $\mathscr{O}(\text{polylog} \frac{1}{\varepsilon})$ for a target accuracy of $\varepsilon$ when the dimension $d\geq 2$. This improvement results from the Hamiltonian simulation strategy applied to the Schrödingerized preconditioning dynamics, coupled with the smoothing of initial data in the extended space. |
| title | Quantum preconditioning method for linear systems problems via Schrödingerization |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.06866 |