  The recent developments of quantum computing present potential novel pathways
for quantum chemistry, as the increased computational power of quantum
computers could be harnessed to naturally encode and solve electronic structure
problems. Theoretically exact quantum algorithms for chemistry have been
proposed (e.g. Quantum Phase Estimation) but the limited capabilities of
current noisy intermediate scale quantum devices (NISQ) motivated the
development of less demanding hybrid algorithms. In this context, the
Variational Quantum Eigensolver (VQE) algorithm was successfully introduced as
an effective method to compute the ground state energy of small molecules. The
current study investigates the Folded Spectrum (FS) method as an extension to
the VQE algorithm for the computation of molecular excited states. It provides
the possibility of directly computing excited states around a selected target
energy, using the same ansatz as for the ground state calculation. Inspired by
the variance-based methods from the Quantum Monte Carlo literature, the FS
method minimizes the energy variance, thus requiring a computationally
expensive squared Hamiltonian. We alleviate this potentially poor scaling by
employing a Pauli grouping procedure, identifying sets of commuting Pauli
strings that can be evaluated simultaneously. This allows for a significant
reduction of the computational cost. We apply the FS-VQE method to small
molecules (H$_2$,LiH), obtaining all electronic excited states with chemical
accuracy on ideal quantum simulators.
