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This research investigates the practical limits of a specific quantum algorithm for solving the nonlinear Vlasov equation, concluding that current convergence criteria impose dissipation requirements far exceeding those found in real plasma applications. The study highlights that query and gate complexities for this approach are polynomially worse than classical counterparts due to matrix properties inherent to the Carleman-linearization method. Consequently, the work underscores a significant near-term barrier to using quantum computing for realistic kinetic plasma simulations.
Solving the Nonlinear Vlasov Equation on a Quantum Computer
Quantum 10, 2206 (2026). https://doi.org/10.22331/q-2026-09-10-2206 The 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.
The 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.
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