  We propose an algorithm for simulating the dynamics of a geometrically local
Hamiltonian $A$ under a small geometrically local perturbation $\alpha B$. In
certain regimes, the algorithm achieves the optimal scaling and outperforms the
state-of-the-art algorithms. By moving into the interaction frame of $A$ and
classically computing the Magnus expansion of the interaction-picture
Hamiltonian, our algorithm bypasses the need for ancillary qubits. In analyzing
its performance, we develop a framework to capture the quasi-locality of the
Magnus operators, leading to a tightened bound for the error of the Magnus
truncation. The Lieb-Robinson bound also guarantees the efficiency of computing
the Magnus operators and of their subsequent decomposition into elementary
quantum gates. These features make our algorithm appealing for near-term and
early-fault-tolerant simulations.
