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We present a self consistent field approach (SCF) within the AdaptiveDerivative-Assembled Problem-Tailored Ansatz Variational Quantum Eigensolver(ADAPT-VQE) framework for efficient quantum simulations of chemical systems onnear-term quantum computers. To this end, our ADAPT-VQE-SCF approach combinesthe idea of generating an ansatz with a small number of parameters, resultingin shallow-depth quantum circuits with a direct minimization of an energyexpression which is correct to second order with respect to changes in themolecular orbital basis. Our numerical analysis, including calculations for thetransition metal complex ferrocene (Fe$\rm (C_5H_5)_2$), indicates thatconvergence in the self-consistent orbital optimization loop can be reachedwithout a considerable increase in the number of two-qubit gates in the quantumcircuit by comparison to a VQE optimization in the initial molecular orbitalbasis. Moreover, the orbital optimization can be carried out simultaneouslywithin each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us toimplement a routine analogous to CASSCF, a cornerstone of state-of-the-artcomputational chemistry, in a hardware-efficient manner on near-term quantumcomputers. Hence, ADAPT-VQE-SCF paves the way towards a paradigm shift forquantitative quantum-chemistry simulations on quantum computers by requiringfewer qubits and opening up for the use of large and flexible atomic orbitalbasis sets in contrast to earlier methods that are predominantly based on theidea of full active spaces with minimal basis sets.

A self-consistent field approach for the variational quantum eigensolver: orbital optimization goes adaptive