量子变分方法用于等离子体平衡：Grad-Shafranov方程的代价函数设计

中文摘要 AI 辅助

非线性偏微分方程的求解是变分量子算法（VQAs）的一个重要应用领域。在本工作中，我们将VQAs的应用扩展到等离子体物理，通过求解二维Grad-Shafranov方程来实现，该方程在理想磁流体力学框架下描述了等离子体流体的平衡态。针对这一问题，我们开发并研究了两种不同的代价函数公式，并比较了它们在解保真度、收敛速度和量子资源需求方面的性能。我们的结果来自无噪声模拟，表明代价函数公式的选择显著影响变分算法的性能。Weak和Picard公式表现出互补特性：前者始终收敛更快，而后者实现了略低的最终不保真度。利用这些互补特性，我们引入了一种混合优化策略，该策略结合了Weak公式的快速初始收敛与Picard公式的更高最终精度。此外，这两种方法由于边界条件实现方式的不同，展现出不同的量子资源需求。这些结果凸显了代价函数设计在开发用于非线性偏微分方程的高效VQAs中的重要性。

英文摘要

The solution of nonlinear partial differential equations constitutes an important application domain for variational quantum algorithms (VQAs). In this work, we extend the application of VQAs to plasma physics by solving the two-dimensional Grad-Shafranov equation, which describes the equilibrium state of a plasma fluid within the framework of ideal magnetohydrodynamics. For this problem, we develop and investigate two distinct cost function formulations and compare their performance in terms of solution fidelity, convergence speed and quantum resource requirements. Our results, obtained from noiseless simulations, show that the choice of cost function formulation significantly affects the performance of the variational algorithm. The Weak and Picard formulations exhibit complementary characteristics: the former consistently converges faster, whereas the latter achieves slightly lower final infidelities. Exploiting these complementary properties, we introduce a hybrid optimization strategy that combines the rapid initial convergence of the Weak formulation with the higher final accuracy of the Picard formulation. In addition, the two approaches display different quantum resource requirements due to their distinct implementations of boundary conditions. These results highlight the importance of cost function design in the development of efficient VQAs for nonlinear partial differential equations.

发表机构

• Technical University of Crete（克里特理工大学）
• National University of Singapore（新加坡国立大学）
• NCSR Demokritos（德谟克利特国家科学研究中心）
• University of Southampton（南安普顿大学）

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