发表机构
George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology（乔治·W·伍德鲁夫机械工程学院，佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出量子同伦摄动方法（QHPM），通过保持希尔伯特空间维度并利用变分量子模拟减少量子比特和电路深度，提高非线性PDE求解的可扩展性，并用涡量输运方程等示例验证。
AI 中文摘要
求解非线性偏微分方程（PDE）在众多科学与工程应用中具有重要意义。近年来，量子计算被引入作为一种求解非线性PDE的替代计算范式。本文提出了一种名为量子同伦摄动方法（QHPM）的新方法，旨在通过两个方面提高求解非线性PDE的可扩展性。首先，通过同伦摄动将非线性PDE线性化后，希尔伯特空间的维度保持不变。其次，采用变分量子模拟框架获得解，其中通过函数编码减少了量子比特数量，并降低了参数化电路的深度。本文的额外贡献是引入了新的准则，用于选择同伦级数截断阶数和电路深度，以实现成本效益高的QHPM。所提出的方法通过多个示例进行了演示，包括涡量输运方程和约化磁流体动力学方程。
英文摘要
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.
