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Solving the Nonlinear Vlasov Equation on a Quantum Computer

Tamás Vaszary, Animesh Datta, Tom Goffrey, and Brian Appelbe

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AbstractThe practical applicability of a recent Carleman-linearization-based quantum algorithm for solving ordinary differential equations (ODEs) with quadratic nonlinearities is investigated for the nonlinear electrostatic Vlasov equation with Krook-type collision operators. The equation is discretized on a (1+1)-dimensional phase-space grid and mapped onto the input of the quantum algorithm. Upper bounds for the query and gate complexities are derived in the limit of large grid sizes and found to be polynomially larger than the time complexity of the corresponding classical algorithms, primarily due to the dimension, sparsity, and norm of the Carleman-linearized evolution matrix. The convergence criteria are shown to impose severe restrictions on physically relevant plasma applications, requiring dissipation levels far exceeding those provided by the Krook operator.Featured image: A schematic overview of the procedure used in this work. The left panel shows the classical reformulation of the nonlinear plasma model, from phase-space discretization through Carleman linearization to a linear system. The right panel shows the subsequent quantum algorithm, including state preparation, quantum linear-system solving, and extraction of physical observables.Popular summaryPlasma physics describes the behaviour of ionized matter in settings ranging from astrophysical plasmas to controlled fusion. Simulating plasmas on classical computers is challenging because their dynamics are governed by nonlinear equations over a high-dimensional phase space. Quantum computers offer a potential alternative, but efficiently treating nonlinear dynamics remains an important obstacle. In this work, we investigate a recent quantum algorithm based on Carleman linearization, which transforms a nonlinear differential equation into a larger linear system that can be solved using quantum linear-system methods. We apply this approach to the nonlinear Vlasov equation and analyse both its convergence and computational complexity. We find that the convergence conditions require dissipation much stronger than that provided by physically realistic collision rates. Moreover, for large phase-space grids, the resulting quantum algorithm has polynomially greater time complexity than a straightforward classical solver of the same discretized equations. These results show that the potential for quantum advantage in nonlinear plasma simulation depends on the formulation and algorithmic approach. In the present setting, the convergence and complexity limitations identified here motivate the development of alternative formulations and algorithmic improvements that can operate in more physically relevant regimes.► BibTeX data@article{Vaszary2026solvingnonlinear, doi = {10.22331/q-2026-09-10-2206}, url = {https://doi.org/10.22331/q-2026-09-10-2206}, title = {Solving the {N}onlinear {V}lasov {E}quation on a {Q}uantum {C}omputer}, author = {Vaszary, Tam{\'{a}}s and Datta, Animesh and Goffrey, Tom and Appelbe, Brian}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2206}, month = sep, year = {2026} }► References [1] A. Alekseenko and Craig Euler. A Bhatnagar–Gross–Krook kinetic model with velocity-dependent collision frequency and corrected relaxation of moments. Continuum Mechanics and Thermodynamics, 01 2013. 10.1007/s00161-014-0407-0. https://doi.org/10.1007/s00161-014-0407-0 [2] Abtin Ameri, Erika Ye, Paola Cappellaro, Hari Krovi, and Nuno F. Loureiro. Quantum algorithm for the linear Vlasov equation with collisions. Phys. Rev. A, 107: 062412, Jun 2023. 10.1103/PhysRevA.107.062412. https://doi.org/10.1103/PhysRevA.107.062412 [3] Dong An, Jin-Peng Liu, Daochen Wang, and Qi Zhao.

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A, 106: 032408, Sep 2022a. 10.1103/PhysRevA.106.032408. https://doi.org/10.1103/PhysRevA.106.032408 [65] Julien Zylberman, Giuseppe Di Molfetta, Marc Brachet, Nuno F. Loureiro, and Fabrice Debbasch. Hybrid Quantum-Classical Algorithm for Hydrodynamics. 2022b. 10.48550/arXiv.2202.00918. https://doi.org/10.48550/arXiv.2202.00918 [66] Óscar Amaro and Diogo Cruz. A Living Review of Quantum Computing for Plasma Physics. 2023. 10.48550/arXiv.2302.00001. https://doi.org/10.48550/arXiv.2302.00001Cited byCould not fetch Crossref cited-by data during last attempt 2026-09-10 09:33:31: Could not fetch cited-by data for 10.22331/q-2026-09-10-2206 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-09-10 09:33:38: Cannot retrieve data from ADS due to rate limitations.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. AbstractThe practical applicability of a recent Carleman-linearization-based quantum algorithm for solving ordinary differential equations (ODEs) with quadratic nonlinearities is investigated for the nonlinear electrostatic Vlasov equation with Krook-type collision operators. The equation is discretized on a (1+1)-dimensional phase-space grid and mapped onto the input of the quantum algorithm. Upper bounds for the query and gate complexities are derived in the limit of large grid sizes and found to be polynomially larger than the time complexity of the corresponding classical algorithms, primarily due to the dimension, sparsity, and norm of the Carleman-linearized evolution matrix. The convergence criteria are shown to impose severe restrictions on physically relevant plasma applications, requiring dissipation levels far exceeding those provided by the Krook operator.Featured image: A schematic overview of the procedure used in this work. The left panel shows the classical reformulation of the nonlinear plasma model, from phase-space discretization through Carleman linearization to a linear system. The right panel shows the subsequent quantum algorithm, including state preparation, quantum linear-system solving, and extraction of physical observables.Popular summaryPlasma physics describes the behaviour of ionized matter in settings ranging from astrophysical plasmas to controlled fusion. Simulating plasmas on classical computers is challenging because their dynamics are governed by nonlinear equations over a high-dimensional phase space. Quantum computers offer a potential alternative, but efficiently treating nonlinear dynamics remains an important obstacle. In this work, we investigate a recent quantum algorithm based on Carleman linearization, which transforms a nonlinear differential equation into a larger linear system that can be solved using quantum linear-system methods. We apply this approach to the nonlinear Vlasov equation and analyse both its convergence and computational complexity. We find that the convergence conditions require dissipation much stronger than that provided by physically realistic collision rates. Moreover, for large phase-space grids, the resulting quantum algorithm has polynomially greater time complexity than a straightforward classical solver of the same discretized equations. These results show that the potential for quantum advantage in nonlinear plasma simulation depends on the formulation and algorithmic approach. In the present setting, the convergence and complexity limitations identified here motivate the development of alternative formulations and algorithmic improvements that can operate in more physically relevant regimes.► BibTeX data@article{Vaszary2026solvingnonlinear, doi = {10.22331/q-2026-09-10-2206}, url = {https://doi.org/10.22331/q-2026-09-10-2206}, title = {Solving the {N}onlinear {V}lasov {E}quation on a {Q}uantum {C}omputer}, author = {Vaszary, Tam{\'{a}}s and Datta, Animesh and Goffrey, Tom and Appelbe, Brian}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2206}, month = sep, year = {2026} }► References [1] A. Alekseenko and Craig Euler. A Bhatnagar–Gross–Krook kinetic model with velocity-dependent collision frequency and corrected relaxation of moments. Continuum Mechanics and Thermodynamics, 01 2013. 10.1007/s00161-014-0407-0. https://doi.org/10.1007/s00161-014-0407-0 [2] Abtin Ameri, Erika Ye, Paola Cappellaro, Hari Krovi, and Nuno F. Loureiro. Quantum algorithm for the linear Vlasov equation with collisions. Phys. Rev. A, 107: 062412, Jun 2023. 10.1103/PhysRevA.107.062412. https://doi.org/10.1103/PhysRevA.107.062412 [3] Dong An, Jin-Peng Liu, Daochen Wang, and Qi Zhao.

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