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Parameterization of Tensor Network Contraction

Author Bryan O'Gorman

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Bryan O'Gorman
  • Berkeley Quantum Information & Computation Center, University of California, Berkeley, CA, USA
  • Quantum Artificial Intelligence Laboratory, NASA Ames, Moffett Field, CA, USA


The author thanks Benjamin Villalonga for motivating this work, useful discussions, and feedback on the manuscript.

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Bryan O'Gorman. Parameterization of Tensor Network Contraction. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We present a conceptually clear and algorithmically useful framework for parameterizing the costs of tensor network contraction. Our framework is completely general, applying to tensor networks with arbitrary bond dimensions, open legs, and hyperedges. The fundamental objects of our framework are rooted and unrooted contraction trees, which represent classes of contraction orders. Properties of a contraction tree correspond directly and precisely to the time and space costs of tensor network contraction. The properties of rooted contraction trees give the costs of parallelized contraction algorithms. We show how contraction trees relate to existing tree-like objects in the graph theory literature, bringing to bear a wide range of graph algorithms and tools to tensor network contraction. Independent of tensor networks, we show that the edge congestion of a graph is almost equal to the branchwidth of its line graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Quantum information theory
  • Theory of computation → Graph algorithms analysis
  • tensor networks
  • parameterized complexity
  • tree embedding
  • congestion


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