  We present a low-space overhead simulation algorithm based on the truncated
Dyson series for time-dependent quantum dynamics. This algorithm is applied to
simulating time-independent Hamiltonians by transitioning to the interaction
picture, where some portions are made time-dependent. This can provide a
favorable complexity trade-off as the algorithm scales exponentially better
with derivatives of the time-dependent component than the original Hamiltonian.
We show that this leads to an exponential improvement in gate complexity for
simulating some classes of diagonally dominant Hamiltonian. Additionally we
show that this can reduce the gate-complexity scaling for simulating $N$-site
Hubbard models for time $t$ with arbitrary long-range interactions as well as
reduce the cost of quantum chemistry simulations within a similar-sized
plane-wave basis to $\widetilde{\mathcal{O}}(N^2t)$ from
$\widetilde{\mathcal{O}}(N^{11/3}t)$. We also show a quadratic improvement in
query complexity for simulating sparse time-dependent Hamiltonians, which may
be of independent interest.
