  Quantum simulation of the electronic structure problem is one of the most
researched applications of quantum computing. The majority of quantum
algorithms for this problem encode the wavefunction using $N$ Gaussian
orbitals, leading to Hamiltonians with ${\cal O}(N^4)$ second-quantized terms.
We avoid this overhead and extend methods to the condensed phase by utilizing a
dual form of the plane wave basis which diagonalizes the potential operator,
leading to a Hamiltonian representation with ${\cal O}(N^2)$ second-quantized
terms. Using this representation we can implement single Trotter steps of the
Hamiltonians with linear gate depth on a planar lattice. Properties of the
basis allow us to deploy Trotter and Taylor series based simulations with
respective circuit depths of ${\cal O}(N^{7/2})$ and $\widetilde{\cal
O}(N^{8/3})$ for fixed charge densities - both are large asymptotic
improvements over all prior results. Variational algorithms also require
significantly fewer measurements to find the mean energy in this basis,
ameliorating a primary challenge of that approach. We conclude with a proposal
to simulate the uniform electron gas (jellium) using a low depth variational
ansatz realizable on near-term quantum devices. From these results we identify
simulations of low density jellium as a promising first setting to explore
quantum supremacy in electronic structure.
