  We consider simulating quantum systems on digital quantum computers. We show
that the performance of quantum simulation can be improved by simultaneously
exploiting commutativity of the target Hamiltonian, sparsity of interactions,
and prior knowledge of the initial state. We achieve this using Trotterization
for a class of interacting electrons that encompasses various physical systems,
including the plane-wave-basis electronic structure and the Fermi-Hubbard
model. We estimate the simulation error by taking the transition amplitude of
nested commutators of the Hamiltonian terms within the $\eta$-electron
manifold. We develop multiple techniques for bounding the transition amplitude
and expectation of general fermionic operators, which may be of independent
interest. We show that it suffices to use
$\left(\frac{n^{5/3}}{\eta^{2/3}}+n^{4/3}\eta^{2/3}\right)n^{o(1)}$ gates to
simulate electronic structure in the plane-wave basis with $n$ spin orbitals
and $\eta$ electrons, improving the best previous result in second quantization
up to a negligible factor while outperforming the first-quantized simulation
when $n=\eta^{2-o(1)}$. We also obtain an improvement for simulating the
Fermi-Hubbard model. We construct concrete examples for which our bounds are
almost saturated, giving a nearly tight Trotterization of interacting
electrons.
