  Near-term quantum computers will be limited in the number of qubits on which
they can process information as well as the depth of the circuits that they can
coherently carry out. To-date, experimental demonstrations of algorithms such
as the Variational Quantum Eigensolver (VQE) have been limited to small
molecules using minimal basis sets for this reason. In this work we propose
incorporating an orbital optimization scheme into quantum eigensolvers wherein
a parameterized partial unitary transformation is applied to the basis
functions set in order to reduce the number of qubits required for a given
problem. The optimal transformation is found by minimizing the ground state
energy with respect to this partial unitary matrix. Through numerical
simulations of small molecules up to 16 spin orbitals, we demonstrate that this
method has the ability to greatly extend the capabilities of near-term quantum
computers with regard to the electronic structure problem. We find that VQE
paired with orbital optimization consistently achieves lower ground state
energies than traditional VQE when using the same number of qubits and even
frequently achieves lower ground state energies than VQE methods using more
qubits.
