  We consider simulating quantum systems on digital quantum computers. We show
that the performance of quantum simulation can be improved by simultaneously
exploiting the commutativity of Hamiltonian, the sparsity of interactions, and
the prior knowledge of 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 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
$\mathcal{O}\left(\frac{n^{5/3}}{\eta^{2/3}}+n^{4/3}\eta^{2/3}\right)$ gates to
simulate electronic structure in the plane-wave basis with $n$ spin orbitals
and $\eta$ electrons up to a negligible factor, improving the best previous
result in second quantization while outperforming the first-quantized
simulation when $n=\mathcal{O}\left(\eta^2\right)$. 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.
