A quantum computing algorithm for the investigation of the molecular excited states

Quantum computing is emerging as a new paradigm for the solution of quantum chemistry problems. Recently, the Variational Quantum Eigensolver (VQE) algorithm has been proposed and successfully applied to the simulation of the ground state properties of simple molecular systems in a real quantum device. The calculation of molecular excited state properties constitute an additional challenge for both classical and quantum electronic structure algorithms. In fact, in addition to the calculation of a well-converged ground state wavefunction, one needs to devise schemes for the evaluation of the higher energy states, which - in general - are not accessible through the simple optimization procedure. In this work, a perturbative approach is applied to the ground state wavefunction to derive a pseudo-eigenvalue problem, which size is characterized by a favorable scaling in the number of electrons. The different matrix elements are measured on the quantum hardware using the ground state wavefunction parametrized according to the UCC and the hardware efficient Ansätze described in [Barkoutsos et al., Phys. Rev. A 98, 022322]. The method is applied to the calculation of the excited states of simple molecules, including H2, LiH and H2O.