Towards a polynomial algorithm for optimal contraction sequence of tensor networks from trees.

The computational cost of contracting a tensor network depends on the sequence of contractions, but to decide the sequence of contractions with a minimal computational cost on an arbitrary network has been proved to be an NP-complete problem. In this work, we conjecture that the problem may be a polynomial one if we consider the computational complexity instead. We propose a polynomial algorithm for the optimal contraction complexity of tensor tree network, which is a specific and widely applied network structure. We prove that for any tensor tree network, the proposed algorithm can achieve a sequence of contractions that guarantees the minimal time complexity and a linear space complexity simultaneously. To illustrate the validity of our idea, numerical simulations are presented that evidence the significant benefits when the network scale becomes large. This work will have great potential for the efficient processing of various physical simulations and pave the way for the further exploration of the computational complexity of tensor contraction on arbitrary tensor networks.