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

Time-dependent Hamiltonian simulation with $L^1$-norm scaling