AI 中文总结
该研究提出一种基于水平集公式的非线性标量守恒律量子算法，经薛定谔化等处理实现显式门级，在高维可观测量估计上具量子优势，数值实验验证了其有效性。
AI 中文摘要
非线性偏微分方程的量子算法仍具挑战性，因为非线性动力学无法直接适配幺正量子模拟。基于水平集公式，我们构建了一种用于求解标量守恒律的量子算法，并提供了显式门级实现。该非线性方程首先被提升为线性刘维尔方程，通过有限差分法离散，再经薛定谔化嵌入幺正演化。我们进一步开发了从演化态估计相关可观测量的量子程序，为完整算法建立了误差界和门复杂度估计。复杂度对比表明，在空间维度足够高时，于态制备和预言机访问的标准假设下，可观测量估计存在量子优势。最后，数值实验验证了所提方法的准确性、多维适用性及预测的缩放性。
英文摘要
Quantum algorithms for nonlinear partial differential equations remain challenging because nonlinear dynamics are not directly amenable to unitary quantum simulation. Building on the level-set formulation, we construct a quantum algorithm and provide an explicit gate-level implementation for solving scalar conservation laws. The nonlinear equation is first lifted to a linear Liouville equation, discretized by finite differences, and then embedded into a unitary evolution through Schrödingerisation. We further develop quantum procedures for estimating relevant observables from the evolved state. Error bounds and gate-complexity estimates are established for the complete algorithm. The resulting complexity comparison demonstrates a quantum advantage for observable estimation in sufficiently high spatial dimensions, under standard assumptions on state preparation and oracle access. Finally, numerical experiments validate the accuracy, multidimensional applicability, and predicted scaling of the proposed method.
