李布拉德安方法用于求解非线性偏微分方程
Lindbladian Homotopy Analysis Method to Solve Nonlinear Partial Differential Equations
AI总结：
本文提出一种量子微分方程求解方法，通过将非线性问题转化为递归线性PDE序列，并利用李布拉德安动力学模拟求解，相较于其他方法，其希尔伯特空间维度增长更慢。
摘要：
量子科学计算旨在利用量子计算机解决工程和科学问题，如模拟和优化。求解常微分方程和偏微分方程（PDEs）在模拟中至关重要。然而，现有的量子方法在求解非线性PDEs时面临维度灾难和线性化过程中的收敛问题。本文提出了一种李布拉德安同调分析方法（LHAM）作为量子微分方程求解器，用于模拟非幺正和非线性动力学。原始非线性问题首先通过同调分析方法转换为递归序列的线性PDEs，并重新表述为更高维的低块三角线性齐次自治系统。然后，将解嵌入密度矩阵并通过李布拉德安动力学模拟获得。与其他方法如Carleman线性化和Koopman-von Neumann方法相比，LHAM中希尔伯特空间维度仅随截断误差的倒数以对数方式增长。LHAM通过包含Burgers方程和简化磁流体动力学方程的非线性PDEs进行了演示。
英文摘要：
Quantum scientific computing is to solve engineering and science problems such as simulation and optimization on quantum computers. 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 nonlinear and nonunitary dynamics. The original nonlinear problem is first converted to a recursive sequence of linear PDEs with the homotopy analysis method and reformulated as a higher-dimensional lower block triangular linear homogeneous autonomous system. The solution is then embedded in the density matrix and obtained through the Lindblad dynamics simulation. Compared to other methods such as Carleman linearization and the Koopman-von Neumann approach where the dimension of Hilbert space increases polynomially with the inverse of truncation error, the Hilbert space dimension in LHAM increases only logarithmically. LHAM is demonstrated with nonlinear PDEs including Burgers' equation and reduced magnetohydrodynamics equations.
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