Efficient Application of Tensor Network Operators to Tensor Network States

TL;DR

This paper introduces a new efficient algorithm for applying tensor network operators to tensor network states, significantly improving runtime performance and demonstrating effectiveness in both benchmark and circuit simulation scenarios.

Contribution

The authors develop a Cholesky-based compression method for applying tensor network operators, extending existing techniques to general tree structures with improved efficiency.

Findings

CBC performs at least an order of magnitude faster than existing methods.

Complex tree structures can outperform linear ones in circuit simulations.

CBC maintains high performance across different bond dimensions.

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.