NP-Hardness of Tensor Network Contraction Ordering

TL;DR

This paper investigates the computational complexity of finding optimal contraction sequences in tensor networks, establishing NP-hardness for minimizing time complexity and clarifying relationships between different contraction problems.

Contribution

It proves that the time complexity minimization problem is NP-hard and relates it to existing contraction ordering problems, providing new insights into their computational difficulty.

Findings

CMS is NP-hard.

CMS is easier than OMS in general and in tree cases.

Relationships between hardness of tensor network contraction problems are established.

Abstract

We study the optimal order (or sequence) of contracting a tensor network with a minimal computational cost. We conclude 2 different versions of this optimal sequence: that minimize the operation number (OMS) and that minimize the time complexity (CMS). Existing results only shows that OMS is NP-hard, but no conclusion on CMS problem. In this work, we firstly reduce CMS to CMS-0, which is a sub-problem of CMS with no free indices. Then we prove that CMS is easier than OMS, both in general and in tree cases. Last but not least, we prove that CMS is still NP-hard. Based on our results, we have built up relationships of hardness of different tensor network contraction problems.