Calculation of excited states via symmetry constraints in the Variational Quantum Eigensolver

The variational quantum eigensolver (VQE) requires specification of symmetries that describe the system, e.g. spin state and number of electrons. This opens the possibility of using VQE to obtain excited states. In this paper, various unitary coupled cluster (UCC) ansatz\"{e} applied to excited states are investigated, using quantum circuits to represent single reference and multireference wavefunctions. These ansatz\"{e} include standard UCCSD, as well as recently developed UCCGSD and k-UpCCGSD approaches designed to tackle the calculation of states with strong multireference character. These studies are carried out on H$_2$, H$_3$, and the methylene radical CH$_2$ as examples of molecules with singlet, doublet and triplet ground states respectively, at different spin configurations and charge states. Our calculations are mostly in agreement with results from full configuration interaction, thus showing that the VQE algorithm can calculate the lowest excited state at a certain symmetry by setting constraints in the qubit register encoding the starting mean field state. In the case of the CH$_2$ using k-UpCCGSD, up-shifted singlet states relative to FCI are observed. Comparing the optimized cluster amplitudes between k-UpCCGSD and UCCGSD, the lack of generalized double excitations in k-UpCCGSD accounts for the increased energy of the first singlet state. In addition, we observe crossover between different spin states when using generalized ansatz\"{e} which can be prevented by penalty terms in the hamiltonian and qubit registers encoding multireference states. Comparison of calculations with qubit registers encoding different types of singlets (closed-shell and open-shell) suggest that conventional VQE may be able to calculate higher excited states beyond the first one for a particular set of symmetries, although with a significant loss in accuracy of the results.