  The quantum simulation of quantum chemistry is a promising application of
quantum computers. However, for N molecular orbitals, the $\mathcal{O}(N^4)$
gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter
steps makes simulation based on such primitives challenging. We substantially
reduce the gate complexity of such primitives through a two-step low-rank
factorization of the Hamiltonian and cluster operator, accompanied by
truncation of small terms. Using truncations that incur errors below chemical
accuracy, we are able to perform Trotter steps of the arbitrary basis
electronic structure Hamiltonian with $\mathcal{O}(N^3)$ gate complexity in
small simulations, which reduces to $\mathcal{O}(N^2 \log N)$ gate complexity
in the asymptotic regime, while our unitary Coupled Cluster Trotter step has
$\mathcal{O}(N^3)$ gate complexity as a function of increasing basis size for a
given molecule. In the case of the Hamiltonian Trotter step, these circuits
have $\mathcal{O}(N^2)$ depth on a linearly connected array, an improvement
over the $\mathcal{O}(N^3)$ scaling assuming no truncation. As a practical
example, we show that a chemically accurate Hamiltonian Trotter step for a 50
qubit molecular simulation can be carried out in the molecular orbital basis
with as few as 4,000 layers of parallel nearest-neighbor two-qubit gates,
consisting of fewer than 100,000 non-Clifford rotations. We also apply our
algorithm to iron-sulfur clusters relevant for elucidating the mode of action
of metalloenzymes.
