State-Specific Orbital Optimization for Enhanced Excited-States Calculation on Quantum Computers

We propose a state-specific orbital optimization scheme for improving the accuracy of excited states of the electronic structure Hamiltonian for the use on near-term quantum computers, which can be combined with any overlap-based excited-state quantum eigensolver. We derived the gradient of the overlap term between different states generated by different orbitals with respect to the orbital rotation matrix and use the gradient-based optimization methods to optimize the orbitals. This scheme allows for more flexibility in the choice of orbitals. We implement the state-specific orbital optimization scheme with the variational quantum deflation (VQD) algorithm, and show that it achieves higher accuracy than the state-averaged orbital optimization scheme on various molecules including H4 and LiH.