  Finding the ground state energy of electrons subject to an external electric
field is a fundamental problem in computational chemistry. We prove that this
electronic-structure problem, when restricted to a fixed single-particle basis
and fixed number of electrons, is QMA-complete. Schuch and Verstraete have
shown hardness for the electronic-structure problem with an additional
site-specific external magnetic field, but without the restriction to a fixed
basis. In their reduction, a local Hamiltonian on qubits is encoded in the
site-specific magnetic field. In our reduction, the local Hamiltonian is
encoded in the choice of spatial orbitals used to discretize the
electronic-structure Hamiltonian. As a step in their proof, Schuch and
Verstraete show a reduction from the antiferromagnetic Heisenberg Hamiltonian
to the Fermi-Hubbard Hamiltonian. We combine this reduction with the fact that
the antiferromagnetic Heisenberg Hamiltonian is QMA-hard to observe that the
Fermi-Hubbard Hamiltonian on generic graphs is QMA-hard, even when all the
hopping coefficients have the same sign. We then reduce from Fermi-Hubbard by
showing that an instance of Fermi-Hubbard can be closely approximated by an
instance of the Electronic-Structure Hamiltonian in a fixed basis. Finally, we
show that estimating the energy of the lowest-energy Slater-determinant state
(i.e., the Hartree-Fock state) is NP-complete for the Electronic-Structure
Hamiltonian in a fixed basis.
