收缩平面张量网络的复杂性
The Complexity of Contracting Planar Tensor Network
AI总结：
本文研究平面图和有限元图上张量网络收缩的指数复杂度，提出基于边分隔器的快速算法，并通过低维张量转换和CP分解加速计算，同时证明#ETH下的次指数时间下界。
摘要：
张量网络在量子计算和机器学习等许多研究领域中一直是一个重要的概念和技术。我们研究在两种特殊图结构上收缩张量网络的指数复杂度：平面图和有限元图。我们证明任何有限元图都有一个大小为 $O(d\sqrt{\max\{Δ,d\}N})$ 的边分隔器。此外，基于平面布尔张量网络收缩的 $2^{O(\sqrt{ΔN})}$ 时间算法，我们开发了一个 $2^{O(d\sqrt{\max\{Δ,d\}N})}$ 时间算法，用于收缩由 $N$ 个布尔张量组成的张量网络，其底层图是一个最大度为 $Δ$ 的有限元图，并且在平面骨架中没有面具有超过 $d$ 条边界边。我们使用两种方法通过将高维张量转换为低维张量来加速指数算法。对于任何维度为 $k$ 的布尔对称张量，我们构造了一个 $O(k)$ 大小的平面小工具，其中该小工具仅由维度不超过 $5$ 的布尔张量组成。另一种方法是根据其 CP 分解将任何张量分解为一系列向量（一元函数）。我们还证明了在计数指数时间假设（#ETH）下，收缩张量网络的次指数时间下界成立。
英文摘要：
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 $O(d\sqrt{\max\{Δ,d\}N})$ size edge separator. Furthermore, we develop a $2^{O(d\sqrt{\max\{Δ,d\}N})}$ time algorithm to contracting a tensor network consisting of $N$ Boolean tensors, whose underlying graph is a finite element graph with maximum degree $Δ$ and has no face with more than $d$ boundary edges in the planar skeleton, based on the $2^{O(\sqrt{ΔN})}$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
