Quantum Molecular Science III

The electronic structure problem presents a computational challenge in quantum chemistry, particularly when scaling to larger molecular systems. Hartree-Fock (HF) methods, by approximating many-body fermionic systems at the mean-field level, offer a feasible classical computational approximation to the electronic ground state. In the current NISQ (Noisy Intermediate-Scale Quantum) era of quantum computers, HF serves a dual role: it provides a compact, free-fermion reference state for molecular systems and often functions as the initial guess for hybrid quantum-classical algorithms, such as the Variational Quantum Eigensolver (VQE). Despite its fundamental role, the limitations of HF in capturing electronic correlations necessitate further investigation into exactly when and why HF performs well, particularly in the context of emerging quantum algorithms.

We introduce a family of permutationally invariant model Hamiltonians, which we call the completely delocalized (CDL) Hamiltonian, for which the mean-field approximation becomes exact due to a fermionic de Finetti theorem. Our goal is to investigate the broader applicability of the CDL Hamiltonians to real molecular systems and assess the corresponding effectiveness of HF in approximating ground states for such systems. We define a metric that quantifies the distance between a given discretized molecular Hamiltonian and this model. This distance reflects how far a molecular system deviates from the idealized permutational symmetry of the CDL Hamiltonian. Our findings indicate that the two-body terms of real molecular Hamiltonians are closer to the CDL manifold when HF energies approach the complete basis set (CBS) limit. This correlation provides evidence that systems that align more closely with the CDL model tend to yield better HF approximations. Conversely, synthetic Hamiltonians with randomized interactions show a rapid increase in their distance to the CDL manifold as the number of fermionic modes grows, highlighting the breakdown of HF we expect in arbitrary electronic systems. These insights extend beyond classical computational chemistry, offering potential improvements in quantum state preparation—an essential step for efficient quantum simulations of molecular systems.