  Finding ground state energies on current quantum processing units (QPUs)
using algorithms like the variational quantum eigensolver (VQE) continues to
pose challenges. Hardware noise severely affects both the expressivity and
trainability of parametrized quantum circuits, limiting them to shallow depths
in practice. Here, we demonstrate that both issues can be addressed by
synergistically integrating VQE with a quantum subspace expansion, allowing for
an optimal balance between quantum and classical computing capabilities and
costs. We perform a systematic benchmark analysis of the iterative
quantum-assisted eigensolver of [K. Bharti and T. Haug, Phys. Rev. A {\bf 104},
L050401 (2021)] in the presence of hardware noise. We determine ground state
energies of 1D and 2D mixed-field Ising spin models on noisy simulators and on
the IBM QPUs ibmq_quito (5 qubits) and ibmq_guadalupe (16 qubits). To maximize
accuracy, we propose a suitable criterion to select the subspace basis vectors
according to the trace of the noisy overlap matrix. Finally, we show how to
systematically approach the exact solution by performing controlled quantum
error mitigation based on probabilistic error reduction on the noisy backend
fake_guadalupe.
