Automatic Structural Search of Tensor Network States including Entanglement Renormalization

TL;DR

This paper introduces an algorithm for automatically searching optimal tensor network structures, including entanglement renormalization, to better represent complex quantum states with non-uniform entanglement patterns, improving accuracy and efficiency.

Contribution

The study presents a novel algorithm for structural search of tensor networks, including ER, based on local structure reconstruction and variational energy minimization, demonstrating its effectiveness on spin models.

Findings

Achieved exact ground energy calculation for spin-$1/2$ tetramer model using MERA.

Improved variational energy, fidelity, and entanglement entropy in random XY models.

Using existing TN structures as preprocessing enhances the algorithm's performance.

Abstract

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.