Structure-preserving quantum algorithms for linear and nonlinear Hamiltonian systems
Hsuan-Cheng Wu, Xiantao Li
TL;DR
The paper develops structure-preserving quantum algorithms for Hamiltonian systems by embedding symplectic integrators into quantum workflows. For linear Hamiltonians, it uses Runge-Kutta Gauss collocation to produce a symplectic map and solves the resulting linear system with a quantum linear system solver, achieving energy preservation and a complexity of $\tilde{O}(T \|K\| \kappa(V)^2)$ up to polylog factors. For nonlinear Hamiltonians, it employs Carleman embedding to transform the dynamics into (approximately) linear, still symplectic evolution on a truncated finite system, and provides complexity bounds $\tilde{O}\left(\frac{T^{1+2\log \kappa(V)}}{\epsilon^{2\log \kappa(V)}} (\|F_1\|+\|F_2\|)\right)$ with an $\epsilon$-approximately symplectic map. The results demonstrate potential quantum advantages for simulating large-scale Hamiltonian systems while preserving essential physical properties, though exact symplectic preservation for nonlinear dynamics remains open and the truncation introduces an exponential dependence on certain parameters under no-resonance constraints. Overall, this work advances structure-preserving quantum simulation by integrating symplectic integrators and Carleman embedding within a quantum-linear-algebra framework, with quantified performance implications.
Abstract
Hamiltonian systems of ordinary and partial differential equations are fundamental mathematical models spanning virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is the underlying 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.