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The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for N molecular orbitals, the 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 allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with O (N 3) gate complexity in small simulations, which reduces to O (N 2) gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with 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 …