Quantum simulation of Helmholtz equations via Schr{ö}dingerization
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911085657849856 |
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| author | Gu, Anjiao Jin, Shi Ma, Chuwen |
| author_facet | Gu, Anjiao Jin, Shi Ma, Chuwen |
| contents | The Helmholtz equation is a prototypical model for time-harmonic wave propagation. Numerical solutions become increasingly challenging as the wave number $k$ grows, due to the equation's elliptic yet noncoercive character and the highly oscillatory nature of its solutions, with wavelengths scaling as $1/k$. These features lead to strong indefiniteness and large system sizes.
We present a quantum algorithm for solving such indefinite problems, built upon the Schrödingerization framework. This approach reformulates linear differential equations into Schrödinger-type systems by capturing the steady state of damped dynamics. A warped phase transformation lifts the original problem to a higher-dimensional formulation, making it compatible with quantum computation. To suppress numerical pollution, the algorithm incorporates asymptotic dispersion correction. It achieves a query complexity of $\mathcal{O}(κ^2\text{polylog}\varepsilon^{-1})$, where $κ$ is the condition number and $\varepsilon$ the desired accuracy. For the Helmholtz equation, a simple preconditioner further reduces the complexity to $\mathcal{O}(κ\text{polylog}\varepsilon^{-1})$. Our constructive extension to the quantum setting is broadly applicable to all indefinite problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23547 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum simulation of Helmholtz equations via Schr{ö}dingerization Gu, Anjiao Jin, Shi Ma, Chuwen Numerical Analysis The Helmholtz equation is a prototypical model for time-harmonic wave propagation. Numerical solutions become increasingly challenging as the wave number $k$ grows, due to the equation's elliptic yet noncoercive character and the highly oscillatory nature of its solutions, with wavelengths scaling as $1/k$. These features lead to strong indefiniteness and large system sizes. We present a quantum algorithm for solving such indefinite problems, built upon the Schrödingerization framework. This approach reformulates linear differential equations into Schrödinger-type systems by capturing the steady state of damped dynamics. A warped phase transformation lifts the original problem to a higher-dimensional formulation, making it compatible with quantum computation. To suppress numerical pollution, the algorithm incorporates asymptotic dispersion correction. It achieves a query complexity of $\mathcal{O}(κ^2\text{polylog}\varepsilon^{-1})$, where $κ$ is the condition number and $\varepsilon$ the desired accuracy. For the Helmholtz equation, a simple preconditioner further reduces the complexity to $\mathcal{O}(κ\text{polylog}\varepsilon^{-1})$. Our constructive extension to the quantum setting is broadly applicable to all indefinite problems. |
| title | Quantum simulation of Helmholtz equations via Schr{ö}dingerization |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.23547 |