Quantum Algorithm for Numerical Energy Gradient Calculations at the Full Configuration Interaction Level of Theory

A Bayesian phase difference estimation (BPDE) algorithm allows us to compute the energy gap of two electronic states of a given Hamiltonian directly by utilizing the quantum superposition of their wave functions. Here we report an extension of the BPDE algorithm to the direct calculation of the energy difference of two molecular geometries. We apply the BPDE algorithm for the calculation of numerical energy gradients based on the two-point finite-difference method, enabling us to execute geometry optimization of one-dimensional molecules at the full-CI level on a quantum computer. Results of numerical quantum circuit simulations of the geometry optimization of the H₂ molecule with the STO-3G and 6-31G basis sets, the LiH and BeH₂ molecules at the full-CI/STO-3G level, and the N₂ molecule at the CASCI(6e,6o)/6-311G* level are given.