On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups

Authors Zhili Chen , Joshua A. Grochow , Youming Qiao , Gang Tang , Chuanqi Zhang

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Zhili Chen
  • Center for Quantum Software and Information, University of Technology Sydney, Australia
Joshua A. Grochow
  • Departments of Computer Science and Mathematics, University of Colorado Boulder, CO, USA
Youming Qiao
  • Center for Quantum Software and Information, University of Technology Sydney, Australia
Gang Tang
  • Center for Quantum Software and Information, University of Technology Sydney, Australia
Chuanqi Zhang
  • Center for Quantum Software and Information, University of Technology Sydney, Australia


We thank the anonymous reviewers for their careful reading and helpful suggestions.

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Zhili Chen, Joshua A. Grochow, Youming Qiao, Gang Tang, and Chuanqi Zhang. On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 31:1-31:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We study the complexity of isomorphism problems for d-way arrays, or tensors, under natural actions by classical groups such as orthogonal, unitary, and symplectic groups. These problems arise naturally in statistical data analysis and quantum information. We study two types of complexity-theoretic questions. First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the complexity of the isomorphism problem over a classical group to that over the general linear group. Second, for a fixed group type (orthogonal, unitary, or symplectic), we compare the complexity of the isomorphism problems for different actions. Our main results are as follows. First, for orthogonal and symplectic groups acting on 3-way arrays, the isomorphism problems reduce to the corresponding problems over the general linear group. Second, for orthogonal and unitary groups, the isomorphism problems of five natural actions on 3-way arrays are polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the 3-tensor isomorphism problem for any fixed d > 3. For unitary groups, the preceding result implies that LOCC classification of tripartite quantum states is at least as difficult as LOCC classification of d-partite quantum states for any d. Lastly, we also show that the graph isomorphism problem reduces to the tensor isomorphism problem over orthogonal and unitary groups.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • complexity class
  • tensor isomorphism
  • polynomial isomorphism
  • group isomorphism
  • local operations and classical communication


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