Truncating loopy tensor networks by zero-mode gauge fixing

TL;DR

The paper tackles the inefficiency of truncating loopy tensor networks by introducing zero-mode truncation via gauge fixing on a cut bond. By identifying zero modes of the local state metric tensor, the method eliminates linear dependence among bond states and reduces the bond dimension while preserving the target state, with f ≈ N_D/|Z_D|^2 as a guiding error. It generalizes to broader zero modes, connects to Environment Assisted Truncation (EAT), and defines a loopiness measure to distinguish loopy from non-loopy bonds. Across diverse 2D quantum models (Ising, Heisenberg, Z2 gauge field, t–J) and TRG applications, the zero-mode truncation (ZMT) consistently yields better initial truncation than standard schemes, enabling more faithful subsequent variational optimization. The results demonstrate that principled initialization based on zero modes improves accuracy and robustness when handling loop-induced entanglement in loopy tensor networks.

Abstract

Loopy tensor networks have internal correlations that often make their compression inefficient. We show that even local bond optimization can make better use of the insight it has locally into relevant loop correlations. By cutting the bond, we define a set of states whose linear dependence can be used to truncate the bond dimension. The linear dependence is eliminated with zero modes of the states' metric tensor. The method is illustrated by a series of examples for the infinite pair entangled projected state (iPEPS) and for the periodic matrix product state (pMPS) that occurs in the tensor renormalization group (TRG) step. In all examples, it provides better initial truncation errors than standard initialization.

Figures (14)

• Figure 1: Cartoon virtual loop entanglement --- The top panel shows a four-tensor plaquette in a larger tensor network (TN). The four tensors are contracted by the black bond indices with dimension $D$. Additionally, there is a blue loop carrying a virtual index $j$ that is decoupled from any physical index. The TN state is a sum over $j=1...d$, $\hbox{$| \rm TN \rangle$}=\sum_{j=1}^d\hbox{$| \psi_j \rangle$}$, where each state $\hbox{$| \psi_j \rangle$}$ is the same and proportional to the TN state. The bottom panel shows the same plaquette after the indices $i$ and $j$ were merged into a single index $k=1...Dd$. Its bond dimension is $d$ times bigger than necessary to represent the TN state. Any single state $\hbox{$| \psi_j \rangle$}$ with the smaller bond dimension $D$ would suffice to represent the same state: $\hbox{$| \rm TN \rangle$}\propto\hbox{$| \psi_j \rangle$}$.
• Figure 2: Bond zero modes. --- In (a) the grey semi-ellipsis contains a tensor network (TN). The red lines are its physical indices. All TN's internal bond indices are hidden except for an index $j$ represented by the blue line. Here the summation over $j$ is explicit. Each value of $j$ defines a state $\hbox{$| \psi_j \rangle$}$. In (b) the bond is cut to define more general states $\hbox{$| \psi_{ij} \rangle$}$. Now the state in (a) can be written as $\hbox{$| \psi \rangle$}=\sum_{i,j=1}^D\delta_{ij}\hbox{$| \psi_{ij} \rangle$}$. In (c) a contraction of $\hbox{$| \psi_{ij} \rangle$}$ with its conjugate defines the metric tensor in \ref{['eq:g_ij']}. In (d) the singular value decomposition \ref{['eq:UlambdaV']} is absorbed into the TN to define the new states in \ref{['eq:tilde_psi']}. In (e) a conventional gauge transformation, inserting $S^{-1}S$ in the bond, defines the new states in \ref{['eq:psi^S']}.
• Figure 3: Environment assisted truncation (EAT). --- The metric tensor in Fig. \ref{['fig:psi_j']} (c) is singular value decomposed between its left and right indices and then, in \ref{['eq:g_eat']}, approximated by truncation to the leading singular value. The tensors $g_{L,R}$ can be made Hermitian and non-negative. By definition, a tensor network appears non-loopy to the bond when this approximation is exact.
• Figure 4: Neighborhood tensor update (NTU). In (a) the infinite PEPS (iPEPS) tensor network with two sublattice tensors $a$ and $b$. In (b) left, a two-site Trotter gate is applied to a pair of tensors. The gate's rank is $r$ and its application increases the bond dimension from $D$ to $rD$. In (b) right, the dimension is truncated back to $D$. The initial error of the truncation is the Frobenius norm of the difference between the left and the right. After the initialization, the two tensors $a'$ and $b'$ on the right are further optimized variationally to minimize the error.
• Figure 5: Quantum Ising model - sudden quench. The initial truncation errors and the final error in function of time after the sudden quench. The inset shows loopiness in function of time. Here the iPEPS bond dimension $D=8$ and the time step $dt=0.01$.