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.

翻译：张量网络已成为量子计算和机器学习等多个研究领域的重要概念与技术手段。本文研究了两种特殊图结构——平面图和有限元图上张量网络收缩的指数复杂度。我们证明任意有限元图均存在大小为$O(d\sqrt{\max\{\Delta,d\}N})$的边分离器。进一步地，基于平面布尔张量网络收缩的$2^{O(\sqrt{\Delta N})}$时间算法\cite{fastcounting}，我们提出了一个$2^{O(d\sqrt{\max\{\Delta,d\}N})}$时间算法，用于收缩由$N$个布尔张量构成的张量网络，其底层图为最大度为$\Delta$且平面骨架中无面具有超过$d$条边界边的有限元图。采用两种方法通过将高维张量转化为低维张量来加速指数算法：首先为任意$k$维布尔对称张量构建大小为$O(k)$的平面小工具，该小工具仅由维数不超过5的布尔张量组成；其次根据张量的\emph{CP分解}\cite{tensor-rank}将任意张量分解为一系列向量（一元函数）。我们同时证明了在计数\emph{指数时间假说}(\#ETH)成立的前提下，张量网络收缩的次指数时间下界。