On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness

Authors Joshua A. Grochow , Youming Qiao

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Joshua A. Grochow
  • Departments of Computer Science and Mathematics, University of Colorado, Boulder, CO, USA
Youming Qiao
  • Centre for Quantum Software and Information, University of Technology Sydney, Australia


We would like to thanks James B. Wilson for related discussions, and Uriya First, Lek-Heng Lim, and J. M. Landsberg for help searching for references asking whether dTI could reduce to 3TI. We also thank Nengkun Yu, Yinan Li, and Graeme Smith for explaining the notion of SLOCC, and Ryan Williams for pointing out the problem of distinguishing between ETH and #ETH.

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Joshua A. Grochow and Youming Qiao. On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study the complexity of isomorphism problems for tensors, groups, and polynomials. These problems have been studied in multivariate cryptography, machine learning, quantum information, and computational group theory. We show that these problems are all polynomial-time equivalent, creating bridges between problems traditionally studied in myriad research areas. This prompts us to define the complexity class TI, namely problems that reduce to the Tensor Isomorphism (TI) problem in polynomial time. Our main technical result is a polynomial-time reduction from d-tensor isomorphism to 3-tensor isomorphism. In the context of quantum information, this result gives multipartite-to-tripartite entanglement transformation procedure, that preserves equivalence under stochastic local operations and classical communication (SLOCC).

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Computing methodologies → Linear algebra algorithms
  • Hardware → Quantum communication and cryptography
  • complexity class
  • tensor isomorphism
  • polynomial isomorphism
  • group isomorphism
  • stochastic local operations and classical communication


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