Before revision:  We show that it suffices to use \mathcal{O \left(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} \eta^2  . We also obtain an improvement for simulating the Fermi-Hubbard model.
After revision:  We show that it suffices to use \mathcal{O \left(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} \eta^2 \eta^{2-o(1) . We also obtain an improvement for simulating the Fermi-Hubbard model.