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Finding the ground-state energy of molecules is an important and challengingcomputational problem for which quantum computing can potentially findefficient solutions. The variational quantum eigensolver (VQE) is a quantumalgorithm that tackles the molecular groundstate problem and is regarded as oneof the flagships of quantum computing. Yet, to date, only very small moleculeswere computed via VQE, due to high noise levels in current quantum devices.Here we present an alternative variational quantum scheme that requiressignificantly less qubits. The reduction in qubit number allows for shallowercircuits to be sufficient, rendering the method more resistant to noise. Theproposed algorithm, termed variational quantumselected-configuration-interaction (VQ-SCI), is based on: (a) representing thetarget groundstate as a superposition of Slater determinant configurations,encoded directly upon the quantum computational basis states; and (b) selectinga-priory only the most dominant configurations. This is demonstrated through aset of groundstate calculations of the H$_2$, LiH, BeH$_2$, H$_2$O, NH$_3$ andC$_2$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 chemicalaccuracy using the lowest number of qubits reported to date. Moreover, when theSCI matrix is generated ``on the fly", the VQ-SCI requires exponentially lessmemory than classical SCI methods. This offers a potential remedy to a severememory bottleneck problem in classical SCI calculations. Finally, the proposedscheme is general and can be straightforwardly applied for finding thegroundstate of any Hermitian matrix, outside the chemical context.

A Qubit-Efficient Variational Selected Configuration-Interaction Method