  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.
