  Calculations of molecular spectral properties, like photodissociation rates
and absorption bands, rely on knowledge of the excited state energies of the
molecule of interest. Protocols based on the variational quantum eigensolver
(VQE) are promising candidates to calculate such energies on emerging noisy
intermediate scale quantum (NISQ) computers. The successful implementation of
these protocols on NISQ computers, relies on ans\"atze that can accurately
approximate the molecular states and that can be implemented by shallow quantum
circuits. In this paper, we introduce the excited qubit-excitation-based
adaptive (e-QEB-ADAPT)-VQE protocol to calculate molecular excited state
energies. The e-QEB-ADAPT-VQE constructs efficient problem-tailored ans\"atze
by iteratively appending evolutions of qubit excitation operators. The
e-QEB-ADAPT-VQE is an adaptation of the QEB-ADAPT-VQE protocol, which is
designed to be independent on the choice of an initial reference state. We
perform classical numerical simulations for LiH and BeH$_2$ to benchmark the
performance of the e-QEB-ADAPT-VQE. We demonstrate that the e-QEB-ADAPT-VQE can
construct highly accurate ans\"atze that require at least an order of magnitude
fewer $CNOT$s than standard fixed UCC ans\"atze, such as the UCCSD and the
GUCCSD.
