  Within the evolving domain of quantum computational chemistry, the
Variational Quantum Eigensolver (VQE) has been developed to explore not only
the ground state but also the excited states of molecules. In this study, we
compare the performance of Variational Quantum Deflation (VQD) and
Subspace-Search Variational Quantum Eigensolver (SSVQE) methods in determining
the low-lying excited states of $LiH$. Our investigation reveals that while VQD
exhibits a slight advantage in accuracy, SSVQE stands out for its efficiency,
allowing the determination of all low-lying excited states through a single
parameter optimization procedure. We further evaluate the effectiveness of
optimizers, including Gradient Descent (GD), Quantum Natural Gradient (QNG),
and Adam optimizer, in obtaining $LiH$'s first excited state, with the Adam
optimizer demonstrating superior efficiency in requiring the fewest iterations.
Moreover, we propose a novel approach combining Folded Spectrum VQE (FS-VQE)
with either VQD or SSVQE, enabling the exploration of highly excited states. We
test the new approaches for finding all three $H_4$'s excited states. Folded
Spectrum SSVQE (FS-SSVQE) can find all three highly excited states near $-1.0$
Ha with only one optimizing procedure, but the procedure converges slowly. In
contrast, although Folded spectrum VQD (FS-VQD) gets highly excited states with
individual optimizing procedures, the optimizing procedure converges faster.
