  Finding the ground-state energy of molecules is an important and challenging
computational problem for which quantum computing can potentially find
efficient solutions. The variational quantum eigensolver (VQE) is a quantum
algorithm that tackles the molecular groundstate problem and is regarded as one
of the flagships of quantum computing. Yet, to date, only very small molecules
were computed via VQE, due to high noise levels in current quantum devices.
Here we present an alternative variational quantum scheme that requires
significantly less qubits. The reduction in qubit number allows for shallower
circuits to be sufficient, rendering the method more resistant to noise. The
proposed algorithm, termed variational quantum
selected-configuration-interaction (VQ-SCI), is based on: (a) representing the
target groundstate as a superposition of Slater determinant configurations,
encoded directly upon the quantum computational basis states; and (b) selecting
a-priory only the most dominant configurations. This is demonstrated through a
set of groundstate calculations of the H$_2$, LiH, BeH$_2$, H$_2$O, NH$_3$ and
C$_2$H$_4$ molecules in the sto-3g basis set, performed on IBM quantum devices.
We show that the VQ-SCI reaches the full-CI (FCI) energy within chemical
accuracy using the lowest number of qubits reported to date. Moreover, when the
SCI matrix is generated ``on the fly", the VQ-SCI requires exponentially less
memory than classical SCI methods. This offers a potential remedy to a severe
memory bottleneck problem in classical SCI calculations. Finally, the proposed
scheme is general and can be straightforwardly applied for finding the
groundstate of any Hermitian matrix, outside the chemical context.
