Quantum algorithm for downfolding quantum chemistry Hamiltonians

We develop a quantum computing approach to construct an effective Hamiltonian acting on the reduced subspace of orbitals starting from the parent electronic Hamiltonian that acts upon the complete active space. The effective Hamiltonian is constructed via systematically downfolding the core/virtual orbitals and counting inwards towards the orbitals in the energy neighbourhood of the HOMO-LUMO orbitals. The downfolding is carried out by a sequence of similarity transformations that are block-encoded via the qubitization algorithm in the quantum circuit. The parameters of the similarity transformation are optimised by the variational quantum-classical hybrid algorithm, by choosing an appropriate cost function. 

As a key result of our analysis, we benchmark the energy spectrum obtained from the full-quantum eigensolver (FQE) on the effective Hamiltonian against that obtained from full configuration interaction calculations for three molecular systems: $H_{2}$, $ ext{LiH}$, $ ext{H}_{2} ext{O}$. The energies from both approaches agree to sub-milliHartree precision.

We also go over how to use this problem-solving method to calculate the electronic potential energy surface as a function of nuclei coordinates, as well as the associated force-fields for ab-initio molecular dynamics and Monte-Carlo simulation.