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arXiv:2604.18924

Lindbladian Homotopy Analysis Method to Solve Nonlinear Partial Differential Equations

Eunsik Choi, Jungin E. Kim, Xueling Lu, Yan Wang

math.NAquant-phSubmitted Apr 21, 2026• Announced Apr 22, 2026• 0 citations
v1 · Apr 21, 2026v2 · Apr 26, 2026v3 · Sep 22, 2026
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Abstract

Quantum scientific computing is to solve engineering and science problems such as simulation and optimization on quantum computerss. Solving ordinary and partial differential equations (PDEs) is essential in simulations. However, existing quantum approaches to solve nonlinear PDEs suffer from the issues of curse of dimensionality and convergence during the linearization process. In this paper, a Lindbladian homotopy analysis method (LHAM) is proposed as a quantum differential equation solver to simulate nonunitary and nonlinear dynamics. The original nonlinear problem is first converted to a recursive sequence of linear PDEs with the homotopy analysis and reformulated as a higher-dimensional lower block triangular linear homogeneous system. The solution is then embedded in the density matrix and obtained through the Lindbladian dynamics simulation. Compared to other methods such as the Carleman linearization and Koopman-von Neumann approach where the dimension of Hilbert space increases polynomially with the inverse of truncation error, the Hilbert space in LHAM increases only logarithmically. LHAM is demonstrated nonlinear PDEs including Burgers' equation and magnetohydrodynamics equations.

Comments
Keywords: quantum computing, homotopy analysis method, Lindblad master equation, nonlinear PDE

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Hamiltonian simulation is a fundamental algorithm in quantum computing that has attracted considerable interest owing to its potential to efficiently solve the governing equations of large-scale classical systems. Exponential speedup through Hamiltonian simulation has been rigorously demonstrated in the case of coupled harmonic oscillators. The question arises as to whether Hamiltonian simulations in other physical systems also accelerate exponentially. Schrödingerization is a technique that transforms the governing equations of classical systems into the Schrödinger equation. However, since the Schrödinger equation is a linear equation, Hamiltonian simulation is often limited to linear equations. The research on Hamiltonian simulation methods for nonlinear governing equations remains relatively limited. In this study, we propose a Hamiltonian simulation method for nonlinear partial differential equations (PDEs). The proposed method is named Carleman linearization + Schrödingerization (CLS), which combines Carleman linearization (CL) and warped phase transformation (WPT). CL is first applied to transform a nonlinear PDE into a linear differential equation. This linearized equation is then mapped to the Schrödinger equation via WPT. The original nonlinear PDE can be solved efficiently by the Hamiltonian simulation of the resulting Schrödinger equation. By applying this method, we transform the original governing equation into the Schrödinger equation. Solving the transformed Schrödinger equation then enables the analysis of the original nonlinear equation. As a specific application, we apply this method to the nonlinear reaction-diffusion equation to demonstrate that Hamiltonian simulations are applicable to nonlinear PDEs.
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