  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.
