Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm

TL;DR

This paper presents a flexible, efficient density matrix algorithm for approximating tensor network contractions, improving accuracy and efficiency by incorporating larger environments and low-rank approximations.

Contribution

It introduces a novel method combining low-rank approximations with a density matrix approach to efficiently contract arbitrary tensor networks with high accuracy.

Findings

Outperforms previous algorithms in accuracy

Achieves higher efficiency in tensor network contraction

Effective for networks with complex graph structures

Abstract

Tensor network contractions are widely used in statistical physics, quantum computing, and computer science. We introduce a method to efficiently approximate tensor network contractions using low-rank approximations, where each intermediate tensor generated during the contractions is approximated as a low-rank binary tree tensor network. The proposed algorithm has the flexibility to incorporate a large portion of the environment when performing low-rank approximations, which can lead to high accuracy for a given rank. Here, the environment refers to the remaining set of tensors in the network, and low-rank approximations with larger environments can generally provide higher accuracy. For contracting tensor networks defined on lattices, the proposed algorithm can be viewed as a generalization of the standard boundary-based algorithms. In addition, the algorithm includes a cost-efficient density matrix algorithm for approximating a tensor network with a general graph structure into a tree structure, whose computational cost is asymptotically upper-bounded by that of the standard algorithm that uses canonicalization. Experimental results indicate that the proposed technique outperforms previously proposed approximate tensor network contraction algorithms for multiple problems in terms of both accuracy and efficiency.