Quantum algorithms for general nonlinear dynamics based on the Carleman embedding
- 1PsiQuantum, 700 Hansen Way, Palo Alto, CA 94304, USA
- 2Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Abstract
Important 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 < 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.
Related
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BibTeX
@article{jennings2026quantum,
title = {{Quantum algorithms for general nonlinear dynamics based on the Carleman embedding}},
author = {David Jennings and Kamil Korzekwa and Matteo Lostaglio and Andrew T Sornborger and Yigit Subasi and Guoming Wang},
journal = {Quantum},
year = {2026},
month = oct,
doi = {10.22331/q-2026-10-08-2233},
url = {https://quantum-journal.org/papers/q-2026-10-08-2233/},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Quantum algorithms for general nonlinear dynamics based on the Carleman embedding
AU - David Jennings
AU - Kamil Korzekwa
AU - Matteo Lostaglio
AU - Andrew T Sornborger
AU - Yigit Subasi
AU - Guoming Wang
PY - 2026
DA - 2026/10/08
JO - Quantum
DO - 10.22331/q-2026-10-08-2233
UR - https://quantum-journal.org/papers/q-2026-10-08-2233/
AB - Important 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 < 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.
SN - 2521-327X
ER -