  In this paper, we introduce a tensor network (TN) scheme into the
entanglement augmentation process of the synergistic optimization framework by
Rudolph et al. [arXiv:2208.13673] to build its process systematically for
inhomogeneous systems. Our synergistic approach first embeds the variational
optimal solution of the TN state with the entropic area law, which can be
perfectly optimized in conventional (classical) computers, in a quantum
variational circuit ansatz inspired by the TN state with the entropic volume
law. Next, the framework performs a variational quantum eigensolver (VQE)
process with embedded states as the initial state. We applied the synergistic
to the ground-state analysis of the all-to-all coupled random transverse-field
Ising, XYZ, Heisenberg model, employing the binary multiscale entanglement
renormalization ansatz (MERA) state and branching MERA states as TN states with
entropic area law and volume law, respectively. We then show that the
synergistic accelerates VQE calculations in the three models without an initial
parameter guess of the branching-MERA-inspired ansatz and can avoid a local
solution trapped by a standard VQE with the ansatz in the Ising model. The
improvement of optimizers for MERA in all-to-all coupled inhomogeneous systems,
enhancement, and potential synergistic applications are also discussed.
