  The quantum phase estimation algorithm stands as the primary method for
determining the ground state energy of a molecular electronic Hamiltonian on a
quantum computer. In this context, the ability to initialize a classically
tractable state that has a strong overlap with the desired ground state is
critical as it directly affects the runtime of the algorithm. However, several
numerical studies have shown that this overlap decays exponentially with system
size. In this work, we demonstrate that this decay can be alleviated by
optimizing the molecular orbital basis, for an initial state constructed from a
single Slater determinant. We propose a practical method to achieve this
optimization without knowledge of the true molecular ground state and test this
method numerically. By comparing the resulting optimized orbitals to the
natural orbitals, we find improved overlap. Specifically, for four iron-sulfur
molecules, which are known to suffer from the mentioned decay, we show that our
method yields one to two orders of magnitude improvement compared to localized
molecular orbitals.
