Improving accuracy of tree-tensor network approach by optimization of network structure

TL;DR

This work addresses how the accuracy of tree tensor network (TTN) calculations for quantum many-body states depends on the network structure. It implements and tests a least-EE principle–based TTN structural optimization, including stochastic updating, on two models: the random XY-exchange model under random fields and the Richardson model. For the random XY model, stochastic local reconnections yield the best variational energies largely independently of the initial TTN, while for the Richardson model, non-stochastic optimization with an appropriate scheme can achieve near-exact energies, but stochastic optimization can trap the TTN in unsuitable structures. The results emphasize that choosing the updating scheme and the initial TTN is model-dependent, and suggest practical strategies such as probing multiple small-$\chi$ runs to identify robust TTN structures and then refining with larger $\chi$, with potential extensions to fast optimization of high-rank tensor representations.

Abstract

Numerical methods based on tensor networks have been extensively explored in the research of quantum many-body systems in recent years. It has been recognized that the ability of tensor networks to describe a quantum many-body state crucially depends on the spatial structure of the network. In the previous work [Hikihara et al., Phys. Rev. Res. 5, 013031 (2023)], we proposed an algorithm based on tree tensor networks (TTNs) that automatically optimizes the structure of TTN according to the spatial profile of entanglement in the state of interest. In this paper, we apply the algorithm to the random XY-exchange model under random magnetic fields and the Richardson model in order to analyze how the performance of the algorithm depends on the detailed updating schemes of the structural optimization. We then find that for the random XY model, on the one hand, the algorithm achieves improved accuracy, and the stochastic algorithm, which selects the local network structure probabilistically, is notably effective. For the Richardson model, on the other hand, the resulting numerical accuracy subtly depends on the initial TTN and the updating schemes. In particular, the algorithm without the stochastic updating scheme certainly improves the accuracy, while the one with the stochastic updates results in poor accuracy due to the effect of randomizing the network structure at the early stage of the calculation. These results indicate that the algorithm successfully improves the accuracy of the numerical calculations for quantum many-body states, while it is essential to appropriately choose the updating scheme as well as the initial TTN structure, depending on the systems treated.

Figures (17)

• Figure 1: Examples of TTN. (a) Matrix-Product Network (MPN), (b) Perfect-Binary Tree (PBT), (c) TTN of a generic form, and (d) the same TTN as that in (c). Open circles represent isometries and the solid diamond represents the singular-value matrix. Circles with colors represent bare spins. In (a), (b), and (c), the bare spins are arranged in the order of the site index. Note that within the category of MPN and PBT, there are several networks corresponding to different site orderings.
• Figure 2: (a) Isometry $V_{a,b}^c$ with the legs $a$, $b$, and $c$. (b) Schematic picture of the orthonormal condition Eq. (\ref{['eq:orthonormal']}). Connected legs (bonds) represent a contraction of their degrees of freedom. The double arrow on the right-hand side of (b) represents a Kronecker delta $\delta_{cc'}$.
• Figure 3: (a) Central area at the beginning of a step. (b) Effective ground-state wave function $\Psi(abcd)$ obtained by the diagonalization of the effective Hamiltonian $\tilde{\mathcal{H}}$. (c), (d), and (e) show the possible local connections of the singular-value decompositions Eqs. (\ref{['eq:SVD_ab_cd']}), (\ref{['eq:SVD_ac_bd']}), and (\ref{['eq:SVD_ad_bc']}), respectively. Open circles represent isometries and a solid diamond denotes the singular-value matrix at the canonical center.
• Figure 4: Random averages of the relative reduction in the variational energy, $[\delta r(\chi)]$, for the calculations where the initial TTN is (a) MPN, (b) PBT, and (c) tSDRG-TTN. "NoOpt" represents the calculation without structural optimization, whereas "bondEE", "rDMwgt", and "Stoch" represent respectively the optimization using the bond EE, that using the truncated rDM weight, and the stochastic optimization. In (a), the data of the calculation without structural optimization for MPN are shown in the inset.
• Figure 5: Random averages of the relative reduction in the variational energy, $[\delta r(\chi)]$, at $\chi=40$. Circles, squares, and triangles represent the results for the calculations where the initial TTN is MPN, PBT, and tSDRG-TTN, respectively. "NoOpt" represents the calculation without structural optimization, whereas "bondEE", "rDMwgt", and "Stoch" represent respectively the optimization using the bond EE, that using the truncated rDM weight, and the stochastic optimization. For the calculations with structural optimization, the results for $\chi_{\rm opt}=16$ ($32$) are shown by solid (open) symbols.