  We present a quantum algorithm for the dynamical simulation of time-dependent
Hamiltonians. Our method involves expanding the interaction-picture Hamiltonian
as a sum of generalized permutations, which leads to an integral-free Dyson
series of the time-evolution operator. Under this representation, we perform a
quantum simulation for the time-evolution operator by means of the linear
combination of unitaries technique. We optimize the time steps of the evolution
based on the Hamiltonian's dynamical characteristics, leading to a gate count
that scales with an $L^1$-norm-like scaling with respect only to the norm of
the interaction Hamiltonian, rather than that of the total Hamiltonian. We
demonstrate that the cost of the algorithm is independent of the Hamiltonian's
frequencies, implying its advantage for systems with highly oscillating
components, and for time-decaying systems the cost does not scale with the
total evolution time asymptotically. In addition, our algorithm retains the
near optimal $\log(1/\epsilon)/\log\log(1/\epsilon)$ scaling with simulation
error $\epsilon$.
