Variational Quantum Homotopy Perturbation Method to Solve Nonlinear Partial Differential Equations
Abstract
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.
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BibTeX
@misc{kim2026variational,
title = {{Variational Quantum Homotopy Perturbation Method to Solve Nonlinear Partial Differential Equations}},
author = {Jungin E. Kim and Eunsik Choi and Yan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06357},
eprint = {2609.06357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06357},
}
RIS
TY - UNPB
TI - Variational Quantum Homotopy Perturbation Method to Solve Nonlinear Partial Differential Equations
AU - Jungin E. Kim
AU - Eunsik Choi
AU - Yan Wang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.06357
UR - https://arxiv.org/abs/2609.06357
AB - 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.
ER -