Efficient Application of Tensor Network Operators to Tensor Network States

TL;DR

This work tackles the challenge of efficiently applying tree tensor network operators to tree tensor network states by introducing Cholesky Based Compression (CBC), a density-matrix-inspired, memory-efficient method applicable to both tensor trains and general TTNs. CBC avoids constructing large G tensors by manipulating M directly and employs left-right sweeps to produce compressed TTN representations with controlled bond dimensions. Through extensive benchmarking against Zip-Up, SRC, and DM-based methods on random TTNS/TTNOs and on circuit-like TTN topologies, CBC achieves comparable accuracy to the state-of-the-art while delivering substantial runtime gains in many regimes, particularly as TTN complexity grows. The results show that appropriately structured TTN topologies can outperform linear TTN structures in realistic circuit simulations, and that CBC remains robust across varying bond-dimension settings, making it a strong default choice for TTNO–TTNS contractions in quantum simulations.

Abstract

The performance of tensor network methods has seen constant improvements over the last few years. We add to this effort by introducing a new algorithm that efficiently applies tree tensor network operators to tree tensor network states inspired by the density matrix method and the Cholesky decomposition. This application procedure is a common subroutine in tensor network methods. We explicitly include the special case of tensor train structures and demonstrate how to extend methods commonly used in this context to general tree structures. We compare our newly developed method with the existing ones in a benchmark scenario with random tensor network states and operators. We find our Cholesky-based compression (CBC) performs equivalently to the current state-of-the-art method, while outperforming most established methods by at least an order of magnitude in runtime. We then apply our knowledge to perform circuit simulation of tree-like circuits, in order to test our method in a more realistic scenario. Here, we find that more complex tree structures can outperform simple linear structures and achieve lower errors than those possible with the simple structures. Additionally, our CBC still performs among the most successful methods, showing less dependence on the different bond dimensions of the operator.

Figures (8)

• Figure 1: Diagrammatic representation of the different tree tensor networks. The dashed lines denote a trivial leg, i.e., one of dimension $1$. The upper two tensor networks are tensor-trains, while the lower two are T3NS. The two TTNs on the left represent quantum states, i.e., they are TTNS, while the two on the right represent quantum operators, i.e., TTNOs. The labels on the latter's legs were omitted for visual clarity.
• Figure 2: The different steps of the CBC for a tensor train structure. a) The contraction \ref{['eq:tt_cbc_first_contraction']} and subsequent truncation, b) the contraction \ref{['eq:right_sweep_init_contr']} and subsequent QR-decomposition \ref{['eq:tt_cbc_qr_decomp']} to obtain the new local tensor, c) the final contraction \ref{['eq:tt_cbc_final_contraction']} to find the new projected subsystem tensor used for the next site. Note that the yellow tensors are not used in later steps and can be deleted.
• Figure 3: The different steps of the CBC for a T3NS structure. a) The contraction and truncation from leafs to root \ref{['eq:cbc_tree_first_contraction']}, b) the contraction and truncation from the root to the leafs \ref{['eq:cbc_tree_second_contraction']}, c) The contraction \ref{['eq:cbc_tree_third_contraction']} and subsequent QR-decomposition \ref{['eq:cbc_tree_qr']} to obtain the new local tensor, d) the final contraction \ref{['eq:cbc_final_contraction']} to obtain the new projected subsystem tensor for the parent site. Note that the yellow tensors are not used in later steps and can be deleted.
• Figure 4: The three additional tree structures explored in \ref{['sec:rand_numerics']}. a) T-Tree structure, b) binary tree structure, c) the fork tensor product structure. Here, only the TTNSs are depicted. The respective TTNOs have the exact same structure but with two physical legs per node.
• Figure 5: Error versus runtime plots for the different tree structures, the results for our CBC method are shown in green. Each data point on a line corresponds to the results for one target bond dimension. In the case of the MPS, the bond dimensions range from $2$ to $40$, while for the other structures, they range from $2$ to $10$, increasing by $2$ with each consecutive data point in both cases.