Estimation of multiple molecular electronic energies and degeneracies with QFAMES

Ground state energy estimation is a core task of molecular electronic structure. However, in order to describe chemical phenomena such as reactions and spectroscopy, we also require knowledge about excited states. The Quantum Multiple Eigenvalue Gaussian filtered Search (QMEGS) [Quantum 8, 1487 (2024)] has been proposed as an early fault-tolerant method for estimating multiple eigenvalues simultaneously. QMEGS uses a single trial state, which must have significant overlap with all modes of interest; in practice this can be quite difficult to achieve. Assembling separate target guesses for each eigenstate into a single trial state can however lead to costly state preparation requirements that are ill-suited to the many repetitions required by QMEGS. The recent extension to Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra (QFAMES) [arXiv:2510.07439] has been presented as a method that allows for multiple eigenvalue estimation using instead a set of multiple trial states. This also enables the evaluation of eigenvalue multiplicity and expectation values of observables. Here we explore the application of QFAMES to molecular systems using different collections of trial states and assess the detection of both exact and accidental energy degeneracies. We also evaluate expectation values of chemical interest, such as total spin, for some cases and show that together the results could be useful for estimating energetic ordering in dense manifolds of electronic spin eigenstates.