  The difficulty of simulating quantum dynamics depends on the norm of the
Hamiltonian. When the Hamiltonian varies with time, the simulation complexity
should only depend on this quantity instantaneously. We develop quantum
simulation algorithms that exploit this intuition. For sparse Hamiltonian
simulation, the gate complexity scales with the $L^1$ norm
$\int_{0}^{t}\mathrm{d}\tau\left\lVert H(\tau)\right\lVert_{\max}$, whereas the
best previous results scale with $t\max_{\tau\in[0,t]}\left\lVert
H(\tau)\right\lVert_{\max}$. We also show analogous results for Hamiltonians
that are linear combinations of unitaries. Our approaches thus provide an
improvement over previous simulation algorithms that can be substantial when
the Hamiltonian varies significantly. We introduce two new techniques: a
classical sampler of time-dependent Hamiltonians and a rescaling principle for
the Schr\"{o}dinger equation. The rescaled Dyson-series algorithm is nearly
optimal with respect to all parameters of interest, whereas the sampling-based
approach is easier to realize for near-term simulation. These algorithms could
potentially be applied to semi-classical simulations of scattering processes in
quantum chemistry.
