Two algorithms for excited-states quantum solvers: Theory and application to EOM-UCCSD

Near-term quantum devices promise to revolutionize quantum chemistry, but simulations with the current noisy intermediate-scale quantum (NISQ) devices are not practical due to their high susceptibility to errors. This motivated the design of NISQ algorithms that leverage classical and quantum resources. While several developments have shown promising results for ground-state simulations, extending the algorithms to excited states remains challenging. This paper presents two cost-efficient excited-state algorithms inspired by the classic Davidson algorithm. We implemented the Davidson method into the quantum version of the equation-of-motion unitary coupled-cluster (qEOM-UCC) excited-state method adapted for quantum hardware. The circuit strategies to generate desired excited states are discussed, implemented, and tested. We demonstrate the performance and accuracy of the proposed algorithms (qEOM-UCC/Davidson and its variational variant) by simulations of H2, H4, LiH, and H2O molecules. Similarly to the classic Davidson scheme, the qEOM-UCC/Davidson algorithms are capable of targeting a small number of excited states of the desired character.