Hamiltonian simulation becomes more challenging as the underlying unitary becomes more oscillatory. In such cases, an algorithm with commutator scaling and a weak dependence, such as logarithmic, on the derivatives of the Hamiltonian is desired. We introduce a new time-dependent Hamiltonian simulation algorithm based on the Magnus series expansion that exhibits both features. Importantly, when applied to unbounded Hamiltonian simulation in the interaction picture, we prove that the commutator in the second-order algorithm leads to a surprising fourth-order superconvergence, with an error preconstant independent of the number of spatial grids. This extends the qHOP algorithm [An, Fang, Lin, Quantum 2022] based on first-order Magnus expansion, and the proof of superconvergence is based on semiclassical analysis that is of independent interest.

翻译：当所涉酉演化具有强振荡特性时，含时哈密顿量模拟面临更大挑战。在此类情形中，需要一种兼具对易子缩放特性且对哈密顿量导数具有弱依赖性（如对数依赖）的算法。我们提出了一种基于马格努斯级数展开的新型含时哈密顿量模拟算法，该算法同时具备上述两大特征。关键的是，当将其应用于相互作用表象中的无界哈密顿量模拟时，我们证明了二阶算法中的对易子项可产生令人惊讶的四阶超收敛性，其误差预常数与空间网格数量无关。该工作推广了基于一阶马格努斯展开的qHOP算法[An, Fang, Lin, Quantum 2022]，而超收敛性的证明基于具有独立价值的半经典分析。