Schrodingerization based quantum algorithms for the time-fractional heat equation
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| Format: | Preprint |
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2025
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| _version_ | 1866915510445146112 |
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| author | Jin, Shi Liu, Nana Yu, Yue |
| author_facet | Jin, Shi Liu, Nana Yu, Yue |
| contents | We develop a quantum algorithm for solving high-dimensional time-fractional heat equations. By applying the dimension extension technique from [FKW23], the $d+1$-dimensional time-fractional equation is reformulated as a local partial differential equation in $d+2$ dimensions. Through discretization along both the extended and spatial domains, a stable system of ordinary differential equations is obtained by a simple change of variables. We propose a quantum algorithm for the resulting semi-discrete problem using the Schrodingerization approach from [JLY24a,JLY23,JL24a]. The Schrodingerization technique transforms general linear partial and ordinary differential equations into Schrodinger-type systems--with unitary evolution, making them suitable for quantum simulation. This is accomplished via the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of this method and conduct a comprehensive complexity analysis, demonstrating up to exponential advantage--with respect to the inverse of the mesh size in high dimensions~--~compared to its classical counterparts. Specifically, to compute the solution to time $T$, while the classical method requires at least $\mathcal{O}(N_t d h^{-(d+0.5)})$ matrix-vector multiplications, where $N_t $ is the number of time steps (which is, for example, $\mathcal{O}(Tdh^{-2})$ for the forward Euler method), our quantum algorithms requires $\widetilde{\mathcal{O}}(T^2d^4 h^{-8})$ queries to the block-encoding input models, with the quantum complexity being independent of the dimension $d$ in terms of the inverse mesh size $h^{-1}$. Numerical experiments are performed to validate our formulation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_17713 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schrodingerization based quantum algorithms for the time-fractional heat equation Jin, Shi Liu, Nana Yu, Yue Numerical Analysis We develop a quantum algorithm for solving high-dimensional time-fractional heat equations. By applying the dimension extension technique from [FKW23], the $d+1$-dimensional time-fractional equation is reformulated as a local partial differential equation in $d+2$ dimensions. Through discretization along both the extended and spatial domains, a stable system of ordinary differential equations is obtained by a simple change of variables. We propose a quantum algorithm for the resulting semi-discrete problem using the Schrodingerization approach from [JLY24a,JLY23,JL24a]. The Schrodingerization technique transforms general linear partial and ordinary differential equations into Schrodinger-type systems--with unitary evolution, making them suitable for quantum simulation. This is accomplished via the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of this method and conduct a comprehensive complexity analysis, demonstrating up to exponential advantage--with respect to the inverse of the mesh size in high dimensions~--~compared to its classical counterparts. Specifically, to compute the solution to time $T$, while the classical method requires at least $\mathcal{O}(N_t d h^{-(d+0.5)})$ matrix-vector multiplications, where $N_t $ is the number of time steps (which is, for example, $\mathcal{O}(Tdh^{-2})$ for the forward Euler method), our quantum algorithms requires $\widetilde{\mathcal{O}}(T^2d^4 h^{-8})$ queries to the block-encoding input models, with the quantum complexity being independent of the dimension $d$ in terms of the inverse mesh size $h^{-1}$. Numerical experiments are performed to validate our formulation. |
| title | Schrodingerization based quantum algorithms for the time-fractional heat equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2509.17713 |