  In many approximate approaches to fermionic quantum many-body systems, such
as Hartree-Fock and density functional theory, solving a system of
non-interacting fermions coupled to some effective potential is the
computational bottleneck. In this paper, we demonstrate that this crucial
computational step can be accelerated using recently developed methods for
Gaussian fermionic matrix product states (GFMPS). As an example, we study the
generalized Hartree-Fock method, which unifies Hartree-Fock and self-consistent
BCS theory, applied to Hubbard models with an inhomogeneous potential. We
demonstrate that for quasi-one-dimensional systems with local interactions, our
approach scales approximately linearly in the length of the system while
yielding a similar accuracy to standard approaches that scale cubically in the
system size.
