Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation
Abstract Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. (in: Proceedings of the National Academy of Sciences 118(35), 2021) demonstrated the first efficient quantum algorithm for dissipative quadrat...
| Main Authors: | , , , , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2023
|
| Online Access: | https://hdl.handle.net/1721.1/152927 |
| _version_ | 1811071556756963328 |
|---|---|
| author | Liu, Jin-Peng An, Dong Fang, Di Wang, Jiasu Low, Guang H. Jordan, Stephen |
| author2 | Massachusetts Institute of Technology. Center for Theoretical Physics |
| author_facet | Massachusetts Institute of Technology. Center for Theoretical Physics Liu, Jin-Peng An, Dong Fang, Di Wang, Jiasu Low, Guang H. Jordan, Stephen |
| author_sort | Liu, Jin-Peng |
| collection | MIT |
| description | Abstract
Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. (in: Proceedings of the National Academy of Sciences 118(35), 2021) demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition
$$R < 1$$
R
<
1
, where R measures the ratio of nonlinearity to dissipation using the
$$\ell _2$$
ℓ
2
norm. Here we develop an efficient quantum algorithm based on Liu et al. (2021) for reaction–diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in Liu et al. (2021) to obtain a faster convergence rate under the condition
$$R_D < 1$$
R
D
<
1
, where
$$R_D$$
R
D
measures the ratio of nonlinearity to dissipation using the
$$\ell _{\infty }$$
ℓ
∞
norm. Since
$$R_D$$
R
D
is independent of the number of spatial grid points n while R increases with n, the criterion
$$R_D<1$$
R
D
<
1
is significantly milder than
$$R<1$$
R
<
1
for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information. |
| first_indexed | 2024-09-23T08:53:02Z |
| format | Article |
| id | mit-1721.1/152927 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T08:53:02Z |
| publishDate | 2023 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1529272024-02-05T18:58:09Z Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation Liu, Jin-Peng An, Dong Fang, Di Wang, Jiasu Low, Guang H. Jordan, Stephen Massachusetts Institute of Technology. Center for Theoretical Physics Abstract Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. (in: Proceedings of the National Academy of Sciences 118(35), 2021) demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition $$R < 1$$ R < 1 , where R measures the ratio of nonlinearity to dissipation using the $$\ell _2$$ ℓ 2 norm. Here we develop an efficient quantum algorithm based on Liu et al. (2021) for reaction–diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in Liu et al. (2021) to obtain a faster convergence rate under the condition $$R_D < 1$$ R D < 1 , where $$R_D$$ R D measures the ratio of nonlinearity to dissipation using the $$\ell _{\infty }$$ ℓ ∞ norm. Since $$R_D$$ R D is independent of the number of spatial grid points n while R increases with n, the criterion $$R_D<1$$ R D < 1 is significantly milder than $$R<1$$ R < 1 for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information. 2023-11-09T15:07:09Z 2023-11-09T15:07:09Z 2023-10-31 2023-11-09T04:21:26Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/152927 Liu, Jin-Peng, An, Dong, Fang, Di, Wang, Jiasu, Low, Guang H. et al. 2023. "Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation." en https://doi.org/10.1007/s00220-023-04857-9 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Liu, Jin-Peng An, Dong Fang, Di Wang, Jiasu Low, Guang H. Jordan, Stephen Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation |
| title | Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation |
| title_full | Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation |
| title_fullStr | Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation |
| title_full_unstemmed | Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation |
| title_short | Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation |
| title_sort | efficient quantum algorithm for nonlinear reaction diffusion equations and energy estimation |
| url | https://hdl.handle.net/1721.1/152927 |
| work_keys_str_mv | AT liujinpeng efficientquantumalgorithmfornonlinearreactiondiffusionequationsandenergyestimation AT andong efficientquantumalgorithmfornonlinearreactiondiffusionequationsandenergyestimation AT fangdi efficientquantumalgorithmfornonlinearreactiondiffusionequationsandenergyestimation AT wangjiasu efficientquantumalgorithmfornonlinearreactiondiffusionequationsandenergyestimation AT lowguangh efficientquantumalgorithmfornonlinearreactiondiffusionequationsandenergyestimation AT jordanstephen efficientquantumalgorithmfornonlinearreactiondiffusionequationsandenergyestimation |