Optimized contraction scheme for tensor-network states

In the tensor-network framework, the expectation values of two-dimensional quantum states are evaluated by contracting a double-layer tensor network constructed from initial and final tensor-network states. The computational cost of carrying out this contraction is generally very high, which limits the largest bond dimension of tensor-network states that can be accurately studied to a relatively small value. We propose an optimized contraction scheme to solve this problem by mapping the double-layer tensor network onto an intersected single-layer tensor network. This reduces greatly the bond dimensions of local tensors to be contracted and improves dramatically the efficiency and accuracy of the evaluation of expectation values of tensor-network states. It almost doubles the largest bond dimension of tensor-network states whose physical properties can be efficiently and reliably calculated, and it extends significantly the application scope of tensor-network methods.

Figure 1

(a) PEPS wave function, $|\mathrm{\Psi }\rangle$, defined on the honeycomb lattice. $A_{x_i{y}_i{z}_i}$ and $B_{x_j{y}_j{z}_j}$ are the local tensors defined on sublattices $A$ and $B$, respectively. The dangling bond, $m_i$, represents a physical basis state at site $i$. $x_i$, $y_i$, and $z_i$ are the bond variables of dimension $D$ linking site $i$ along the $x$, $y$, and $z$ directions, respectively. (b) Double-layer tensor-network representation of $\langle \mathrm{\Psi }|\mathrm{\Psi }\rangle$. The red and green layers, linked by black physical bonds, represent $|\mathrm{\Psi }\rangle$ and $\langle \mathrm{\Psi }|$, respectively. (c) The reduced single-layer tensor network, $\langle \mathrm{\Psi }|\mathrm{\Psi }\rangle$, whose bond dimensions are all equal to $D^2$, obtained from (b) by tracing out all physical bonds.

Figure 2

Graphical representation of an RTN whose local tensors are defined by Eqs. (3) and (4) (top) and the boundary MPS defined by Eq. (5) (bottom). Dashed lines separate the transfer matrices used in the iTEBD calculation. $W_{\alpha \beta }\left[{\sigma }_i\right]$ is the local tensor of the boundary MPS.

Figure 3

(a) Nested single-layer mapping of the double-layer tensor-network state shown in Fig. 1 done by shifting the top layer by half a unit cell along the horizontal direction and then compressing it to the bottom layer. (b) Virtual swap tensor $P$, which is defined as the direct product of two Kronecker delta functions. (c) Resulting NTN (top) and graphical representation of the boundary MPS defined by Eq. (7) (bottom). The transfer matrix $T$ is the product of six subtransfer matrices $T_i$ ($i=1,\cdots ,6)$.

Figure 4

Magnetization $M$ of the spin-2 simplex solid state on a Kagome lattice obtained using the NTN method.

Figure 5

Bond dimension $\chi$ dependence of (a) the energy $E$ and (b) the magnetization $M$ obtained by the NTN (blue) and RTN (red) methods for the 3-PESS ground-state wave function of the $S=1/2$ Kagome Heisenberg model. The energy and magnetization can go either up or down with increasing $\chi$, since the expectation value calculations calculated with either the NTN or the RTN are not variational.

Figure 6

(a) Energy $E$ and (b) magnetization $M$ versus $\chi$ for the 3-PESS ground state of the Kagome antiferromagnetic Heisenberg model with $D=24$ obtained with the NTN method. Insets: Exponential and linear fits to the small–$1/\chi$ data for $E$ and $M$, respectively.