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arXiv:2411.06759 · Science China Physics Mechanics and Astronomy

Quantum homotopy analysis method with quantum-compatible linearization for nonlinear partial differential equations

Cheng Xue, Xiao‐Fan Xu, Xi‐Ning Zhuang, Tai‐Ping Sun, Yunjie Wang, Ming-Yang Tan, Chuang-Chao Ye, Huanyu Liu, Yu-Chun Wu, Zhaoyun Chen, Guo‐Ping Guo

Published Jun 18, 2025• 6 citations
arXivHTMLPDF
DOI
10.1007/s11433-024-2584-2

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Solving nonlinear partial differential equations is important in various scientific and engineering applications. Recently, quantum computing was introduced as an alternative computational paradigm for solving nonlinear partial differential equations. In this paper, a new method called the variational quantum homotopy perturbation method is proposed to improve the scalability of solving nonlinear partial differential equations through two aspects. First, the dimension of the Hilbert space remains the same after the nonlinear partial differential equation is linearized through the homotopy perturbation. Second, the solutions are obtained with a variational quantum simulation framework, where the number of qubits is decreased with functional encoding and the depth of parametrized circuits is reduced. The additional contribution of this paper is the introduction of new criteria for selecting the homotopy series truncation order and circuit depth for cost-effective variational quantum homotopy perturbation method. The proposed approach is demonstrated with several examples, including the vorticity transport equation and the reduced magnetohydrodynamics equations.
Nonlinear partial differential equations (PDEs) underpin computational fluid dynamics, yet resolving nonlinear transport on fine grids remains computationally demanding. Quantum linear-evolution algorithms offer a possible route to large-scale simulation but cannot directly propagate nonlinear coupling. Here we develop a Frozen--Oseen quantum homotopy method (FOQHAM) framework that links transport-aware auxiliary-operator selection to the size of the resulting linear representation. We freeze the full Fréchet derivative at a prescribed flow profile, retaining transport and profile-gradient coupling and close each prescribed finite-order homotopy hierarchy as a fixed affine linear system by product lifting. We formulate its propagation through Linear Combinations of Hamiltonian Simulations (LCHS), without outer homotopy iterations or profile updates. For spatially semidiscrete equations, we establish local approximation bounds and exact closure of the finite-order hierarchy. Classical tests on Burgers, Korteweg--de Vries, Zakharov--Kuznetsov, and one- and two-dimensional isothermal compressible Navier--Stokes equations provide numerical evidence of accurate non-iterative approximations in the tested regimes. The framework thus connects nonlinear PDE approximation to quantum linear evolution and offers a conditional path toward quantum acceleration.