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：张量网络如今是量子多体系统和量子电路经典模拟的核心工具。大多数张量方法依赖于最终收缩张量网络以获得结果。尽管收缩操作本身是平凡的，但其执行时间高度依赖于收缩执行的顺序。为此，人们会预先寻找一个最优的收缩执行顺序。然而，存在一个缺陷：寻找最优收缩顺序的一般问题是NP完全的。因此，对于小规模问题（如$n \leq 20$），必须采用指数级算法与启发式方法的混合策略，否则只能寄希望于良好的收缩顺序。为此，以往的研究侧重于后者，试图寻找更好的启发式方法。在本工作中，我们采取更保守的方法，证明树形张量网络可以接受最优线性收缩顺序。除最优性结论外，我们还调整了两种连接排序技术，这些技术可基于我们的工作为任意张量网络保证近似最优顺序。