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Quantum algorithms for general nonlinear dynamics based on the Carleman embedding

David Jennings, Kamil Korzekwa, Matteo Lostaglio, Andrew T Sornborger, Yigit Subasi, and Guoming Wang

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AbstractImportant nonlinear dynamics, such as those found in plasma and fluid systems, are typically hard to simulate on classical computers. Thus, if fault-tolerant quantum computers could efficiently solve such nonlinear problems, it would be a transformative change for many industries. In a recent breakthrough [Liu et al., PNAS 2021], the first efficient quantum algorithm for solving nonlinear differential equations was constructed, based on a single condition $R \lt 1$, where $R$ characterizes the ratio of nonlinearity to dissipation. This result, however, is limited to the class of purely dissipative systems with negative log-norm, which excludes application to many important problems. In this work, we correct technical issues with this and other prior analysis, and substantially extend the scope of nonlinear dynamical systems that can be efficiently simulated on a quantum computer in a number of ways. Firstly, we extend the existing results from purely dissipative systems to a much broader class of stable systems, and show that every quadratic Lyapunov function for the linearized system corresponds to an independent $R$-number criterion for the convergence of the Carleman scheme. Secondly, we extend our stable system results to physically relevant settings where conserved polynomial quantities exist. Finally, we provide extensive results for the class of non-resonant systems. With this, we are able to show that efficient quantum algorithms exist for a much wider class of nonlinear systems than previously known, and prove the BQP-completeness of nonlinear oscillator problems of exponential size. In our analysis, we also obtain several results related to the Poincaré-Dulac theorem and diagonalization of the Carleman matrix, which could be of independent interest.Featured image: Schematic illustration of the Carleman embedding: nonlinear dynamics are represented by a hierarchy of polynomial observables whose evolution is linear in the lifted space. We establish rigorous conditions under which truncated Carleman embeddings accurately approximate the nonlinear dynamics and use these results to derive efficient quantum algorithms.Popular summarySimulating nonlinear dynamical systems, such as those describing fluid flows and plasmas, is central to science and engineering. Their substantial computational cost motivates the search for new approaches, and quantum computing offers a potential way forward. However, quantum operations are inherently linear, so simulating nonlinear behavior requires special embeddings. These represent nonlinear dynamics through a larger set of linear equations that quantum algorithms can process. In this work, we establish new conditions under which the Carleman embedding accurately captures nonlinear dynamics and supports efficient quantum algorithms. This extends previous guarantees for simulating purely dissipative systems on quantum computers to broader classes of stable systems, systems with conserved quantities, and systems exhibiting complex oscillatory behavior. We also identify nonlinear oscillator problems that quantum computers can solve efficiently and that are expected to be intractable for classical computers, providing evidence for exponential quantum advantage. Together, these results strengthen the foundations of quantum simulation and broaden the range of nonlinear phenomena accessible to future quantum computers.► BibTeX data@article{Jennings2026quantumalgorithms, doi = {10.22331/q-2026-10-08-2233}, url = {https://doi.org/10.22331/q-2026-10-08-2233}, title = {Quantum algorithms for general nonlinear dynamics based on the {C}arleman embedding}, author = {Jennings, David and Korzekwa, Kamil and Lostaglio, Matteo and Sornborger, Andrew T and Subasi, Yigit and Wang, Guoming}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2233}, month = oct, year = {2026} }► References [1] Jin-Peng Liu, Herman Øie Kolden, Hari K. Krovi, Nuno F. Loureiro, Konstantina Trivisa, and Andrew M. Childs. ``Efficient quantum algorithm for dissipative nonlinear differential equations''. PNAS 118, e2026805118 (2021). https://doi.org/10.1073/pnas.2026805118 [2] Hari Krovi. ``Improved quantum algorithms for linear and nonlinear differential equations''. Quantum 7, 913 (2023). https://doi.org/10.22331/q-2023-02-02-913 [3] Hsuan-Cheng Wu, Jingyao Wang, and Xiantao Li. ``Quantum algorithms for nonlinear dynamics: Revisiting carleman linearization with no dissipative conditions''. SIAM Journal on Scientific Computing 47, A943–A970 (2025). https://doi.org/10.1137/24M1665799 [4] Lin Lin. ``Lecture notes on quantum algorithms for scientific computation'' (2022). arXiv:2201.08309. arXiv:2201.08309 [5] Dominic W. Berry. ``High-order quantum algorithm for solving linear differential equations''. J. Phys. A 47, 105301 (2014). https://doi.org/10.1088/1751-8113/47/10/105301 [6] Dominic W. Berry, Andrew M. Childs, Aaron Ostrander, and Guoming Wang. ``Quantum algorithm for linear differential equations with exponentially improved dependence on precision''. Commun. Math. Phys. 356, 1057–1081 (2017). https://doi.org/10.1007/s00220-017-3002-y [7] Dominic W. Berry and Pedro C. S. Costa. ``Quantum algorithm for time-dependent differential equations using Dyson series''. Quantum 8, 1369 (2024). https://doi.org/10.22331/q-2024-06-13-1369 [8] Dong An, Jin-Peng Liu, and Lin Lin. ``Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost''. Phys. Rev. Lett. 131, 150603 (2023). https://doi.org/10.1103/PhysRevLett.131.150603 [9] Dong An, Andrew M. Childs, and Lin Lin. ``Quantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters''. Communications in Mathematical Physics 407, 19 (2026). https://doi.org/10.1007/s00220-025-05509-w [10] David Jennings, Matteo Lostaglio, Robert B. Lowrie, Sam Pallister, and Andrew T. Sornborger. ``The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts''. Quantum 8, 1553 (2024). https://doi.org/10.22331/q-2024-12-10-1553 [11] Amit Surana, Abeynaya Gnanasekaran, and Tuhin Sahai. ``An efficient quantum algorithm for simulating polynomial differential equations'' (2022). arXiv:2212.10775. arXiv:2212.10775 [12] Pedro C. S. Costa, Philipp Schleich, Mauro E. S. Morales, and Dominic W. Berry. ``Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling''. npj Quantum Information 11, 141 (2025). https://doi.org/10.1038/s41534-025-01084-z [13] Jin-Peng Liu, Dong An, Di Fang, Jiasu Wang, Guang Hao Low, and Stephen Jordan. ``Efficient quantum algorithm for nonlinear reaction–diffusion equations and energy estimation''. Commun. Math. Phys. 404, 963–1020 (2023). https://doi.org/10.1007/s00220-023-04857-9 [14] Javier Gonzalez-Conde, Dylan Lewis, Sachin S. Bharadwaj, and Mikel Sanz. ``Quantum carleman linearization efficiency in nonlinear fluid dynamics''. Phys. Rev. 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Cambridge University Press. (1999). [65] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. ``Simulating Hamiltonian dynamics with a truncated Taylor series''. Phys. Rev. Lett. 114, 090502 (2015). https://doi.org/10.1103/PhysRevLett.114.090502 [66] Pedro C. S. Costa, Dong An, Yuval R. Sanders, Yuan Su, Ryan Babbush, and Dominic W. Berry. ``Optimal scaling quantum linear-systems solver via discrete adiabatic theorem''. PRX Quantum 3, 040303 (2022). https://doi.org/10.1103/PRXQuantum.3.040303 [67] Alexander M. Dalzell. ``A shortcut to an optimal quantum linear system solver'' (2024). arXiv:2406.12086. arXiv:2406.12086 [68] David Jennings, Matteo Lostaglio, Sam Pallister, Andrew T. Sornborger, and Yiğit Subaşı. ``Randomized adiabatic quantum linear solver algorithm with optimal complexity scaling and detailed running costs''. PRX Quantum 6, 040373 (2025). https://doi.org/10.1103/1xkb-22cc [69] Tosio Kato. ``Perturbation theory for linear operators''. Volume 132. Springer. (2013).Cited by[1] Hari Krovi, "Improved quantum algorithms for linear and nonlinear differential equations", Quantum 7, 913 (2023). [2] Ivan Novikau and Ilon Joseph, "Globalizing the Carleman linear embedding method for nonlinear dynamics", arXiv:2510.15715, (2025). [3] David Jennings, Kamil Korzekwa, Matteo Lostaglio, Richard Ashworth, Emanuele Marsili, and Stephen Rolston, "End-to-End Quantum Algorithm for Nonlinear Fluid Dynamics with Bounded Quantum Advantage", PRX Quantum 7 3, 033060 (2026). [4] David Jennings, Kamil Korzekwa, Matteo Lostaglio, Paul Mannix, Richard Ashworth, Emanuele Marsili, and Stephen Rolston, "Simulating non-trivial incompressible flows with a quantum lattice Boltzmann algorithm", arXiv:2512.05781, (2025). [5] Judd Katz, Gopikrishnan Muraleedharan, and Abhijeet Alase, "Efficient quantum algorithm for solving differential equations with Fourier nonlinearity via Koopman linearization", arXiv:2512.06488, (2025). [6] Reuben Demirdjian, Thomas Hogancamp, and Daniel Gunlycke, "Efficient decomposition of the Carleman linearized Burgers' equation", Physical Review A 113 3, 032408 (2026). [7] Abtin Ameri, Joseph Carolan, Andrew M. Childs, and Hari Krovi, "Quantum lower bounds for simulating fluid dynamics", arXiv:2603.12161, (2026). [8] Reuben Demirdjian, Thomas Hogancamp, Abeynaya Gnanasekaran, Amit Surana, and Daniel Gunlycke, "Quantum Data Loading for Carleman Linearized Systems: Application to the Lattice-Boltzmann Equation", arXiv:2605.00302, (2026). [9] Xiangyu Li, Ahmet Burak Catli, Ho Kiat Lim, Matthew Pocrnic, Dong An, Jin-Peng Liu, and Nathan Wiebe, "Efficient Quantum Simulation for Nonlinear Stochastic Differential Equations", arXiv:2603.12398, (2026). [10] Erik J. Gustafson and Henry Lamm, "Preparing Fermions via Classical Sampling and Linear Combinations of Unitaries", arXiv:2603.22422, (2026). [11] Baoyang Zhang, Dong An, Zhaoyuan Meng, Yefei Yu, Xiaoxiao Xiao, Zhen Lu, and Yue Yang, "Quantum simulation of real-world nonlinear dynamics via Koopman method", arXiv:2607.07338, (2026). [12] Bjorn K. Berntson, David Jennings, Matteo Lostaglio, and Scott Parker, "An end-to-end quantum algorithm for weakly nonlinear plasma physics with superquadratic speedup", arXiv:2607.14308, (2026). [13] Xiantao Li, "A Quantum Path to Partial Differential Equations", arXiv:2607.09639, (2026). [14] Sitanshu Gakkhar, Ala Shayeghi, and David C.

Del Rey Fernández, "Nonlinear semigroups with unbounded generators under Carleman linearization", arXiv:2605.03381, (2026). [15] Hsuan-Cheng Wu and Xiantao Li, "From Nonlinear Stochastic Differential Equations to Quantum Channels: The Kolmogorov--Lindblad Mapping", arXiv:2608.09903, (2026). [16] Matthew Christensen, Tom Goffrey, and Animesh Datta, "Improved Convergence of Carleman-Embedded Quantum Algorithm for the Vlasov-Poisson System", arXiv:2607.28426, (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-10-11 14:48:36). The list may be incomplete as not all publishers provide suitable and complete citation data.On Crossref's cited-by service no data on citing works was found (last attempt 2026-10-11 14:48:34).This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractImportant nonlinear dynamics, such as those found in plasma and fluid systems, are typically hard to simulate on classical computers. Thus, if fault-tolerant quantum computers could efficiently solve such nonlinear problems, it would be a transformative change for many industries. In a recent breakthrough [Liu et al., PNAS 2021], the first efficient quantum algorithm for solving nonlinear differential equations was constructed, based on a single condition $R \lt 1$, where $R$ characterizes the ratio of nonlinearity to dissipation. This result, however, is limited to the class of purely dissipative systems with negative log-norm, which excludes application to many important problems. In this work, we correct technical issues with this and other prior analysis, and substantially extend the scope of nonlinear dynamical systems that can be efficiently simulated on a quantum computer in a number of ways. Firstly, we extend the existing results from purely dissipative systems to a much broader class of stable systems, and show that every quadratic Lyapunov function for the linearized system corresponds to an independent $R$-number criterion for the convergence of the Carleman scheme. Secondly, we extend our stable system results to physically relevant settings where conserved polynomial quantities exist. Finally, we provide extensive results for the class of non-resonant systems. With this, we are able to show that efficient quantum algorithms exist for a much wider class of nonlinear systems than previously known, and prove the BQP-completeness of nonlinear oscillator problems of exponential size. In our analysis, we also obtain several results related to the Poincaré-Dulac theorem and diagonalization of the Carleman matrix, which could be of independent interest.Featured image: Schematic illustration of the Carleman embedding: nonlinear dynamics are represented by a hierarchy of polynomial observables whose evolution is linear in the lifted space. We establish rigorous conditions under which truncated Carleman embeddings accurately approximate the nonlinear dynamics and use these results to derive efficient quantum algorithms.Popular summarySimulating nonlinear dynamical systems, such as those describing fluid flows and plasmas, is central to science and engineering. Their substantial computational cost motivates the search for new approaches, and quantum computing offers a potential way forward. However, quantum operations are inherently linear, so simulating nonlinear behavior requires special embeddings. These represent nonlinear dynamics through a larger set of linear equations that quantum algorithms can process. In this work, we establish new conditions under which the Carleman embedding accurately captures nonlinear dynamics and supports efficient quantum algorithms. This extends previous guarantees for simulating purely dissipative systems on quantum computers to broader classes of stable systems, systems with conserved quantities, and systems exhibiting complex oscillatory behavior. We also identify nonlinear oscillator problems that quantum computers can solve efficiently and that are expected to be intractable for classical computers, providing evidence for exponential quantum advantage. Together, these results strengthen the foundations of quantum simulation and broaden the range of nonlinear phenomena accessible to future quantum computers.► BibTeX data@article{Jennings2026quantumalgorithms, doi = {10.22331/q-2026-10-08-2233}, url = {https://doi.org/10.22331/q-2026-10-08-2233}, title = {Quantum algorithms for general nonlinear dynamics based on the {C}arleman embedding}, author = {Jennings, David and Korzekwa, Kamil and Lostaglio, Matteo and Sornborger, Andrew T and Subasi, Yigit and Wang, Guoming}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2233}, month = oct, year = {2026} }► References [1] Jin-Peng Liu, Herman Øie Kolden, Hari K. Krovi, Nuno F. Loureiro, Konstantina Trivisa, and Andrew M. Childs. ``Efficient quantum algorithm for dissipative nonlinear differential equations''. PNAS 118, e2026805118 (2021). https://doi.org/10.1073/pnas.2026805118 [2] Hari Krovi. ``Improved quantum algorithms for linear and nonlinear differential equations''. Quantum 7, 913 (2023). https://doi.org/10.22331/q-2023-02-02-913 [3] Hsuan-Cheng Wu, Jingyao Wang, and Xiantao Li. ``Quantum algorithms for nonlinear dynamics: Revisiting carleman linearization with no dissipative conditions''. SIAM Journal on Scientific Computing 47, A943–A970 (2025). https://doi.org/10.1137/24M1665799 [4] Lin Lin. ``Lecture notes on quantum algorithms for scientific computation'' (2022). arXiv:2201.08309. arXiv:2201.08309 [5] Dominic W. Berry. ``High-order quantum algorithm for solving linear differential equations''. J. Phys. A 47, 105301 (2014). https://doi.org/10.1088/1751-8113/47/10/105301 [6] Dominic W. Berry, Andrew M. Childs, Aaron Ostrander, and Guoming Wang. ``Quantum algorithm for linear differential equations with exponentially improved dependence on precision''. Commun. Math. Phys. 356, 1057–1081 (2017). https://doi.org/10.1007/s00220-017-3002-y [7] Dominic W. Berry and Pedro C. S. Costa. ``Quantum algorithm for time-dependent differential equations using Dyson series''. Quantum 8, 1369 (2024). https://doi.org/10.22331/q-2024-06-13-1369 [8] Dong An, Jin-Peng Liu, and Lin Lin. ``Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost''. Phys. Rev. Lett. 131, 150603 (2023). https://doi.org/10.1103/PhysRevLett.131.150603 [9] Dong An, Andrew M. Childs, and Lin Lin. ``Quantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters''. Communications in Mathematical Physics 407, 19 (2026). https://doi.org/10.1007/s00220-025-05509-w [10] David Jennings, Matteo Lostaglio, Robert B. Lowrie, Sam Pallister, and Andrew T. Sornborger. ``The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts''. 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Quantum algorithms for general nonlinear dynamics based on the Carleman embedding

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