Tensor networks are nowadays the backbone of classical simulations of quantum many-body systems and quantum circuits. Most tensor methods rely on the fact that we can eventually contract the tensor network to obtain the final result. While the contraction operation itself is trivial, its execution time is highly dependent on the order in which the contractions are performed. To this end, one tries to find beforehand an optimal order in which the contractions should be performed. However, there is a drawback: the general problem of finding the optimal contraction order is NP-complete. Therefore, one must settle for a mixture of exponential algorithms for small problems, e.g., $n \leq 20$, and otherwise hope for good contraction orders. For this reason, previous research has focused on the latter part, trying to find better heuristics. In this work, we take a more conservative approach and show that tree tensor networks accept optimal linear contraction orders. Beyond the optimality results, we adapt two join ordering techniques that can build on our work to guarantee near-optimal orders for arbitrary tensor networks.

翻译：电锯网络如今是量子多体系统和量子电路经典模拟的骨干。 多数高压方法都依赖于我们最终能够将电压网络承包为最终结果这一事实。 虽然收缩操作本身微不足道, 但其执行时间在很大程度上取决于收缩的顺序。 为此, 人们试图事先找到一个最优的收缩顺序。 但是, 存在着一个缺陷: 找到最佳收缩顺序的一般问题是 NP 完成的。 因此, 一个人必须选择一种混合的微量算法, 来对付小问题, 比如 $n\leq 20, 而不是希望得到好收缩订单。 因此, 先前的研究集中在后一部分, 试图找到更好的收缩顺序。 在这项工作中, 我们采取更保守的方法, 并表明树压网络接受最优的线性收缩顺序。 除了最优性的结果外, 我们调整了两种订购技术, 可以在我们的工作基础上, 来保证任意的收缩网络得到近最佳的订单。