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.

翻译：张量网络收缩在统计物理、量子计算和计算机科学领域有着广泛应用。本文提出一种利用低秩近似高效逼近张量网络收缩的方法，其中收缩过程中生成的每个中间张量都被近似为低秩二叉树张量网络。该算法在实施低秩近似时具有灵活纳入大部分环境张量的能力，从而能在给定秩条件下实现高精度。此处的环境指网络中剩余的张量集合，通常采用更大环境范围的低秩近似可获得更高精度。对于定义在晶格上的张量网络收缩，本算法可视为标准边界算法的推广。此外，该算法包含一种计算高效的密度矩阵算法，可将具有一般图结构的张量网络近似转化为树结构，其计算复杂度在渐进上界方面不超过使用正则化的标准算法。实验结果表明，针对多类问题，所提技术在精度和效率方面均优于先前提出的近似张量网络收缩算法。