How an Equi-Ensemble Description Systematically Outperforms the Weighted Ensemble Variational Quantum Eigensolver

Calculating excited states in chemistry is crucial to providing insight into photoinduced molecular behavior beyond the ground state, enabling innovations in spectroscopy, material sciences, and drug design. While several approaches have been developed to compute excited-state properties, finding the best ratio between the computational cost and accuracy remains challenging. The advent of quantum computers brings new perspectives with the development of quantum algorithms that promise an advantage over classical ones. Most of these new algorithms are inspired by previous classical ones but with different pros and cons. In this work, we focus on the choice of the weights within the ensemble variational quantum eigensolver (VQE) based on the generalization of the variational principle for many-body excited states. We compare the performance of the equi-ensemble and weighted ensemble VQE on two types of problems that cover different correlation regimes: (1) the many-body electronic structure problem of formaldimine using the generalized unitary coupled cluster ansatz and (2) the one-body noninteracting Kohn-Sham problem of hydrogen chains using the R_YCNOT hardware-efficient ansatz. While the equi-ensemble objective is to perform a block-diagonalization only, the weighted ensemble has the additional complexity to reach the targeted eigenstates and eigenvalues. In this work, we show that such a difference leads to several significant consequences, such as the increase in circuit depth together with optimization issues. We conclude that one should always favor the use of equi-weights, even if it requires an additional postprocessing step to extract the eigenvalues of the problem.