  Quantum processing units boost entanglement at the level of hardware and
enable physical simulations of highly correlated electron states in molecules
and intermolecular chemical bonds. The variational quantum eigensolver provides
a hardware-efficient toolbox for ground state simulation; however, with
limitations in precision. Even in the absence of noise, the algorithm may
result into a biased energy estimation, particularly with some shallower ansatz
types. Noise additionally degrades entanglement and hinders the ground state
energy estimation (especially if the noise is not fully characterized). Here we
develop a method to exploit the quantum-classical interface provided by
informationally complete measurements to use classical software on top of the
hardware entanglement booster for ansatz- and noise-related error reduction. We
use the tensor network representation of a quantum channel that drives the
noisy state toward the ground one. The tensor network is a completely positive
map by construction, but we elaborate on making the trace preservation
condition local so as to activate the sweeping variational optimization. This
method brings into reach energies below the noiseless ansatz by creating
additional correlations among the qubits and denoising them. Analyzing the
example of the stretched water molecule with a tangible entanglement, we argue
that a hybrid strategy of using the quantum hardware together with the
classical software outperforms a purely classical strategy provided the
classical parts have the same bond dimension. The proposed optimization
algorithm extends the variety of noise mitigation methods and facilitates the
more accurate study of the energy landscape for deformed molecules. The
algorithm can be applied as the final postprocessing step in the quantum
hardware simulation of protein-ligand complexes in the context of drug design.
