Automatic structural optimization of tree tensor networks

The tree tensor network (TTN) provides an essential theoretical framework for the practical simulation of quantum many-body systems, where the network structure defined by the connectivity of the isometry tensors plays a crucial role in improving its approximation accuracy. In this paper, we propose a TTN algorithm that enables us to automatically optimize the network structure by local reconnections of isometries to suppress the bipartite entanglement entropy on their legs. The algorithm can be seamlessly implemented to such a conventional TTN approach as the density-matrix renormalization group. We apply the algorithm to the inhomogeneous antiferromagnetic Heisenberg spin chain, having a hierarchical spatial distribution of the interactions. We then demonstrate that the entanglement structure embedded in the ground state of the system can be efficiently visualized as a perfect binary tree in the optimized TTN. Possible improvements and applications of the algorithm are also discussed.

Figure 1

(a) Schematic picture of the binary tree tensor network (TTN). The ovals (black) and circles (red) represent isometries and bare spin degrees of freedom, respectively. The arrows are the tensor legs, which we call the (auxiliary) bonds. The rhombus (red) represents the singular values. The part surrounded by the dotted curve (red) is the central area. (b) The isometry $V_{ab}^c$. Directions of the bonds are shown by arrows.

Figure 2

A variety of tree tensor network (TTN) geometries. (a) The matrix product network (MPN). (b) The perfect binary TTN. (c) An example of the nonuniform TTN.

Figure 3

Local improvement process. (a) The current central bond $a$ and two isometries connected to the new central bond $e$. (b) The renormalized ground-state wave function ${\tilde{\mathrm{\Psi }}}_{abcd}^{}$. (c) Updated isometries connected to the new central bond $e$.

Figure 4

Three possible applications of singular-value decomposition (SVD) to ${\tilde{\mathrm{\Psi }}}_{abcd}^{}$, as candidates for the local reconnection.

Figure 5

The new central area in the case that Fig. 4 is chosen as the optimal connection and that $c$ is treated as the central bond of the new step. The shaded oval represents $V_{ad}^e$, which has been updated.

Figure 6

Inhomogeneous interactions on the hierarchical chain when (a) $N=2$, (b) $N=4$, and (c) $N=16$.

Figure 7

Optimization process when $\alpha =0.5$ and $N=64$. (a) The initial matrix product network (MPN). (b) After the first sweep. (c) Any time after the second sweep. Only the left half of the tree tensor network (TTN) is presented, while the right half is symmetric with respect to the center of the system.

Figure 8

Nearest neighbor spin correlation function when $\alpha =0.5$ and $N=128$. Open circles and solid squares, respectively, represent the obtained result by the matrix product network (MPN) and the optimal tree tensor network (TTN).

Figure 9

Bond entanglement entropy (EE) calculated when $\alpha =0.5$ and $N=128$ as a function of the horizontal position of the bonds $x$, defined in Appendix C. Open circles represent the EE calculated by the matrix product network (MPN). Other plots are calculated by the optimal tree tensor network (TTN). $y$ denotes the height of the bond in the optimal TTN.

Figure 10

Optimal structures for (a) $\alpha =1.00$, (b) $\alpha =0.80$, and (c) $\alpha =0.75$. The system size is $N=64$. Only the left half of the tree tensor network (TTN) is presented, while the right half is symmetric with respect to the center of the system.

Figure 11

Schematic pictures for the coordinates indicating the positions of the auxiliary bonds for (a) the optimized tree tensor network (TTN) and (b) the matrix product network (MPN). In (a), the coordinates for the bond denoted by a thick arrow (blue) are $(x,y)=(3.5,2)$. In (b), the coordinate for the bond is $x=(3.125+4.0625)/2=3.59375$.