The Complexity of Contracting Planar Tensor Network

TL;DR

This paper investigates the computational complexity of contracting tensor networks on planar and finite element graphs, providing new algorithms and lower bounds that highlight the exponential difficulty of these problems.

Contribution

It introduces a new exponential-time algorithm for contracting tensor networks on finite element graphs and establishes sub-exponential lower bounds based on ETH.

Findings

Developed a $2^{O(d\sqrt{ ext{max}\{ ext{ extDelta},d ext ext{}}N)}}$ time algorithm for tensor contraction.

Proved the existence of a small planar gadget for Boolean symmetric tensors of any dimension.

Established sub-exponential lower bounds for tensor network contraction under ETH.

Abstract

Tensor networks have been an important concept and technique in many research areas, such as quantum computation and machine learning. We study the exponential complexity of contracting tensor networks on two special graph structures: planar graphs and finite element graphs. We prove that any finite element graph has a $O(d\sqrt{\max\{\Delta,d\}N})$ size edge separator. Furthermore, we develop a $2^{O(d\sqrt{\max\{\Delta,d\}N})}$ time algorithm to contracting a tensor network consisting of $N$ Boolean tensors, whose underlying graph is a finite element graph with maximum degree $\Delta$ and has no face with more than $d$ boundary edges in the planar skeleton, based on the $2^{O(\sqrt{\Delta N})}$ time algorithm \cite{fastcounting} for planar Boolean tensor network contractions. We use two methods to accelerate the exponential algorithms by transferring high-dimensional tensors to low-dimensional tensors. We put up a $O(k)$ size planar gadget for any Boolean symmetric tensor of dimension $k$, where the gadget only consists of Boolean tensors with dimension no more than $5$. Another method is decomposing any tensor into a series of vectors (unary functions), according to its \emph{CP decomposition} \cite{tensor-rank}. We also prove the sub-exponential time lower bound for contracting tensor networks under the counting \emph{Exponential Time Hypothesis} (\#ETH) holds.