  We investigate fully self-consistent multiscale quantum-classical algorithms
on current generation superconducting quantum computers, in a unified approach
to tackle the correlated electronic structure of large systems in both quantum
chemistry and condensed matter physics. In both of these contexts, a strongly
correlated quantum region of the extended system is isolated and
self-consistently coupled to its environment via the sampling of reduced
density matrices. We analyze the viability of current generation quantum
devices to provide the required fidelity of these objects for a robust and
efficient optimization of this subspace. We show that with a simple error
mitigation strategy and optimization of compact tensor product bases to
minimize the number of terms to sample, these self-consistent algorithms are
indeed highly robust, even in the presence of significant noises on quantum
hardware. Furthermore, we demonstrate the use of these density matrices for the
sampling of non-energetic properties, including dipole moments and Fermi liquid
parameters in condensed phase systems, achieving a reliable accuracy with
sparse sampling. It appears that uncertainties derived from the iterative
optimization of these subspaces is smaller than variances in the energy for a
single subspace optimization with current quantum hardware. This boosts the
prospect for routine self-consistency to improve the choice of correlated
subspaces in hybrid quantum-classical approaches to electronic structure for
large systems in this multiscale fashion.
