  The quantum simulation of real molecules and materials is one of the most
highly anticipated applications of quantum computing. Algorithms for simulating
electronic structure using a first-quantized plane wave representation are
especially promising due to their asymptotic efficiency. However, previous
proposals for preparing initial states for these simulation algorithms scale
poorly with the size of the basis set. We address this shortcoming by showing
how to efficiently map states defined in a Gaussian type orbital basis to a
plane wave basis with a scaling that is logarithmic in the number of plane
waves. Our key technical result is a proof that molecular orbitals constructed
from Gaussian type basis functions can be compactly represented in a plane wave
basis using matrix product states. While we expect that other approaches could
achieve the same logarithmic scaling with respect to basis set size, our
proposed state preparation technique is also highly efficient in practice. For
example, in a series of numerical experiments on small molecules, we find that
our approach allows us to prepare an approximation to the Hartree-Fock state
using orders of magnitude fewer non-Clifford gates than a naive approach. By
resolving the issue of state preparation, our work allows for the first quantum
simulation of molecular systems whose end-to-end complexity is truly sublinear
in the basis set size.
