Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling
Abstract The solution of large systems of nonlinear differential equations is essential for many applications in science and engineering. We present three improvements to existing quantum algorithms based on the Carleman linearisation technique. First, we use a high-precision method for solving the...
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| Format: | Article |
| Language: | English |
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Nature Portfolio
2025-08-01
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| Series: | npj Quantum Information |
| Online Access: | https://doi.org/10.1038/s41534-025-01084-z |
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| author | Pedro C. S. Costa Philipp Schleich Mauro E. S. Morales Dominic W. Berry |
| author_facet | Pedro C. S. Costa Philipp Schleich Mauro E. S. Morales Dominic W. Berry |
| author_sort | Pedro C. S. Costa |
| collection | DOAJ |
| description | Abstract The solution of large systems of nonlinear differential equations is essential for many applications in science and engineering. We present three improvements to existing quantum algorithms based on the Carleman linearisation technique. First, we use a high-precision method for solving the linearised system that yields logarithmic dependence on the error and near-linear dependence on time. Second, we introduce a rescaling strategy that significantly reduces the cost, which would otherwise scale exponentially with the Carleman order, thus limiting quantum speedups for PDEs. Third, we derive tighter error bounds for Carleman linearisation. We apply our results to a class of discretised reaction-diffusion equations using higher-order finite differences for spatial resolution. We also show that enforcing a stability criterion independent of the discretisation can conflict with rescaling due to the mismatch between the max-norm and the 2-norm. Nonetheless, efficient quantum solutions remain possible when the number of discretisation points is constrained, as enabled by higher-order schemes. |
| format | Article |
| id | doaj-art-990e4d7611b349999b12d6bd9dfb4c06 |
| institution | Kabale University |
| issn | 2056-6387 |
| language | English |
| publishDate | 2025-08-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | npj Quantum Information |
| spelling | doaj-art-990e4d7611b349999b12d6bd9dfb4c062025-08-20T04:03:17ZengNature Portfolionpj Quantum Information2056-63872025-08-0111111410.1038/s41534-025-01084-zFurther improving quantum algorithms for nonlinear differential equations via higher-order methods and rescalingPedro C. S. Costa0Philipp Schleich1Mauro E. S. Morales2Dominic W. Berry3School of Mathematical and Physical Sciences, Macquarie UniversitySchool of Mathematical and Physical Sciences, Macquarie UniversityCentre for Quantum Software and Information, University of Technology SydneySchool of Mathematical and Physical Sciences, Macquarie UniversityAbstract The solution of large systems of nonlinear differential equations is essential for many applications in science and engineering. We present three improvements to existing quantum algorithms based on the Carleman linearisation technique. First, we use a high-precision method for solving the linearised system that yields logarithmic dependence on the error and near-linear dependence on time. Second, we introduce a rescaling strategy that significantly reduces the cost, which would otherwise scale exponentially with the Carleman order, thus limiting quantum speedups for PDEs. Third, we derive tighter error bounds for Carleman linearisation. We apply our results to a class of discretised reaction-diffusion equations using higher-order finite differences for spatial resolution. We also show that enforcing a stability criterion independent of the discretisation can conflict with rescaling due to the mismatch between the max-norm and the 2-norm. Nonetheless, efficient quantum solutions remain possible when the number of discretisation points is constrained, as enabled by higher-order schemes.https://doi.org/10.1038/s41534-025-01084-z |
| spellingShingle | Pedro C. S. Costa Philipp Schleich Mauro E. S. Morales Dominic W. Berry Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling npj Quantum Information |
| title | Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling |
| title_full | Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling |
| title_fullStr | Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling |
| title_full_unstemmed | Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling |
| title_short | Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling |
| title_sort | further improving quantum algorithms for nonlinear differential equations via higher order methods and rescaling |
| url | https://doi.org/10.1038/s41534-025-01084-z |
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