  We present a quantum algorithm for simulating quantum chemistry with gate
complexity $\tilde{O}(N^{1/3} \eta^{8/3})$ where $\eta$ is the number of
electrons and $N$ is the number of plane wave orbitals. In comparison, the most
efficient prior algorithms for simulating electronic structure using plane
waves (which are at least as efficient as algorithms using any other basis)
have complexity $\tilde{O}(N^{8/3} /\eta^{2/3})$. We achieve our scaling in
first quantization by performing simulation in the rotating frame of the
kinetic operator using interaction picture techniques. Our algorithm is far
more efficient than all prior approaches when $N \gg \eta$, as is needed to
suppress discretization error when representing molecules in the plane wave
basis, or when simulating without the Born-Oppenheimer approximation.
