  Tensor network (TN) states, including entanglement renormalization (ER), can
encompass a wider variety of entangled states. When the entanglement structure
of the quantum state of interest is non-uniform in real space, accurately
representing the state with a limited number of degrees of freedom hinges on
appropriately configuring the TN to align with the entanglement pattern.
However, a proposal has yet to show a structural search of ER due to its high
computational cost and the lack of flexibility in its algorithm. In this study,
we conducted an optimal structural search of TN, including ER, based on the
reconstruction of their local structures with respect to variational energy.
Firstly, we demonstrated that our algorithm for the spin-$1/2$ tetramer
singlets model could calculate exact ground energy using the multi-scale
entanglement renormalization ansatz (MERA) structure as an initial TN
structure. Subsequently, we applied our algorithm to the random XY models with
the two initial structures: MERA and the suitable structure underlying the
strong disordered renormalization group. We found that, in both cases, our
algorithm achieves improvements in variational energy, fidelity, and
entanglement entropy. The degree of improvement in these quantities is superior
in the latter case compared to the former, suggesting that utilizing an
existing TN design method as a preprocessing step is important for maximizing
our algorithm's performance.
