Hamiltonian systems of ordinary and partial differential equations are fundamental across modern science and engineering, appearing in models that span virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is the symplectic structure, which preserves geometric properties like phase-space volume over time and energy conservation over an extended period. In this paper, we present quantum algorithms that incorporate symplectic integrators, ensuring the preservation of this key structure. We demonstrate how these algorithms maintain the symplectic properties for both linear and nonlinear Hamiltonian systems. Additionally, we provide a comprehensive theoretical analysis of the computational complexity, showing that our approach offers both accuracy and improved efficiency over classical algorithms. These results highlight the potential application of quantum algorithms for solving large-scale Hamiltonian systems while preserving essential physical properties.

翻译：哈密顿常微分与偏微分方程组是现代科学与工程领域的基石，其模型几乎涵盖所有物理尺度。此类系统中计算方法的鲁棒性与稳定性依赖于一个关键特性——辛结构，该结构能够长时间保持相空间体积等几何性质，并在较长时间尺度上维持能量守恒。本文提出了一类融合辛积分器的量子算法，确保了这一关键结构的保持。我们论证了这些算法在线性与非线性哈密顿系统中如何维持辛特性。此外，我们提供了计算复杂度的完整理论分析，证明该方法在保证精度的同时，相比经典算法具有更高的计算效率。这些成果彰显了量子算法在求解大规模哈密顿系统并保持关键物理特性方面的应用潜力。