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arXiv · 2209.12332

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

Mihail Stoian·Richard Milbradt·Christian B. Mendl
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Abstract

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

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Mihail Stoian, Richard Milbradt, Christian B. Mendl. 2022-09-25. On the Optimal Linear Contraction Order of Tree Tensor Networks, and Beyond. https://doi.org/10.1137/23m161286x

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