  We present a self consistent field approach (SCF) within the Adaptive
Derivative-Assembled Problem-Tailored Ansatz Variational Quantum Eigensolver
(ADAPT-VQE) framework for efficient quantum simulations of chemical systems on
near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines
the idea of generating an ansatz with a small number of parameters, resulting
in shallow-depth quantum circuits with a direct minimization of an energy
expression which is correct to second order with respect to changes in the
molecular orbital basis. Our numerical analysis, including calculations for the
transition metal complex ferrocene (Fe$\rm (C_5H_5)_2$), indicates that
convergence in the self-consistent orbital optimization loop can be reached
without a considerable increase in the number of two-qubit gates in the quantum
circuit by comparison to a VQE optimization in the initial molecular orbital
basis. Moreover, the orbital optimization can be carried out simultaneously
within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to
implement a routine analogous to CASSCF, a cornerstone of state-of-the-art
computational chemistry, in a hardware-efficient manner on near-term quantum
computers. Hence, ADAPT-VQE-SCF paves the way towards a paradigm shift for
quantitative quantum-chemistry simulations on quantum computers by requiring
fewer qubits and opening up for the use of large and flexible atomic orbital
basis sets in contrast to earlier methods that are predominantly based on the
idea of full active spaces with minimal basis sets.
