  A novel parallel hybrid quantum-classical algorithm for the solution of the
quantum-chemical ground-state energy problem on gate-based quantum computers is
presented. This approach is based on the reduced density-matrix functional
theory (RDMFT) formulation of the electronic structure problem. For that
purpose, the density-matrix functional of the full system is decomposed into an
indirectly coupled sum of density-matrix functionals for all its subsystems
using the adaptive cluster approximation to RDMFT. The approximations involved
in the decomposition and the adaptive cluster approximation itself can be
systematically converged to the exact result. The solutions for the
density-matrix functionals of the effective subsystems involves a constrained
minimization over many-particle states that are approximated by parametrized
trial states on the quantum computer similarly to the variational quantum
eigensolver. The independence of the density-matrix functionals of the
effective subsystems introduces a new level of parallelization and allows for
the computational treatment of much larger molecules on a quantum computer with
a given qubit count. In addition, for the proposed algorithm techniques are
presented to reduce the qubit count, the number of quantum programs, as well as
its depth. The new approach is demonstrated for Hubbard-like systems on IBM
quantum computers based on superconducting transmon qubits.
