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The contraction cost of a tensor network depends on the contraction order.However, the optimal contraction ordering problem is known to be NP-hard. Weshow that the linear contraction ordering problem for tree tensor networksadmits a polynomial-time algorithm, by drawing connections to database joinordering. The result relies on the adjacent sequence interchange property ofthe contraction cost, which enables a global decision of the contraction orderbased on local comparisons. Based on that, we specify a modified version of theIKKBZ database join ordering algorithm to find the optimal tree tensor networklinear contraction order. Finally, we extend our algorithm as a heuristic togeneral contraction orders and arbitrary tensor network topologies.

On the Optimal Linear Contraction Order of Tree Tensor Networks, and Beyond