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On the complexity of isomorphism problems for tensors, groups, and polynomials III: actions by classical groups

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

TL;DR

The paper advances the complexity theory of isomorphism problems by studying d-way arrays under classical groups (orthogonal, unitary, symplectic). It shows that, for 3-way tensors, orthogonal and symplectic TI reduce to GL TI, and that for orthogonal and unitary groups the five natural actions on 3-way arrays are polynomial-time equivalent, with d-tensor TI reducible to 3-tensor TI. It also connects these problems to Graph Isomorphism, establishing GI-hardness for certain tensor isomorphism tasks, and provides applications to LOCC classifications in quantum information. The authors develop and adapt techniques such as tensor-systems, the FGS gadget, and path-algebra constructions to handle unitary/orthogonal settings, offering a unified framework and several open questions about TI_G classes and symplectic extensions. Overall, the work broadens the TI landscape beyond GL actions and links tensor isomorphism to classical-group orbit problems with implications for quantum information and combinatorial isomorphism problems.

Abstract

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. Such 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 decision 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 problem 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.

On the complexity of isomorphism problems for tensors, groups, and polynomials III: actions by classical groups

TL;DR

The paper advances the complexity theory of isomorphism problems by studying d-way arrays under classical groups (orthogonal, unitary, symplectic). It shows that, for 3-way tensors, orthogonal and symplectic TI reduce to GL TI, and that for orthogonal and unitary groups the five natural actions on 3-way arrays are polynomial-time equivalent, with d-tensor TI reducible to 3-tensor TI. It also connects these problems to Graph Isomorphism, establishing GI-hardness for certain tensor isomorphism tasks, and provides applications to LOCC classifications in quantum information. The authors develop and adapt techniques such as tensor-systems, the FGS gadget, and path-algebra constructions to handle unitary/orthogonal settings, offering a unified framework and several open questions about TI_G classes and symplectic extensions. Overall, the work broadens the TI landscape beyond GL actions and links tensor isomorphism to classical-group orbit problems with implications for quantum information and combinatorial isomorphism problems.

Abstract

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. Such 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 decision 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 problem 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.
Paper Structure (60 sections, 19 theorems, 23 equations, 3 figures, 1 table)

This paper contains 60 sections, 19 theorems, 23 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of $d$-way arrays and some group actions on them
  3. Motivations for isomorphism problems of $d$-way arrays over classical groups
  4. Warm up: singular value decompositions.
  5. Orthogonal isomorphism of tensors from data analysis.
  6. Unitary isomorphism of tensors from quantum information.
  7. Classical groups arising from Code Equivalence.
  8. Some preliminary remarks on the algorithms for Tensor Isomorphism over classical groups.
  9. Our results
  10. Results on relations between different groups.
  11. Results on relations between different actions.
  12. An application in quantum information.
  13. Overview of the proofs of main results
  14. About Theorem \ref{['thm:graph-iso']}.
  15. About Theorem \ref{['thm:toGL']}.
  16. ...and 45 more sections

Key Result

Theorem 1.3

Suppose a group family $\mathcal{G}=\{\mathcal{G}_n\}$ satisfies that $\mathrm{S}_n\leq \mathcal{G}_n\leq \mathrm{GL}(n, \mathbb{F})$, where here $S_n$ denotes the group of $n \times n$ permutation matrices. Then Graph Isomorphism reduces to Bilinear Form $\mathcal{G}$-Pseudo-isometry, that is, th

Figures (3)

  • Figure 1: Pictorial representation of the reduction for Theorem \ref{['thm:5actions']}; credit for the figure goes to the authors of GQ21, reproduced here with their permission.
  • Figure 2: The bipartite graph encoding a system of three tensors over two $\mathbb{F}$-vector spaces $V_1, V_2$: $T_{1} \in V_{1}^{*}\otimes V_{2}$, $T_{2} \in V_{1} \otimes V_{2}^{*} \otimes V_{2}^{*}$, and $T_{3} \in V_{1} \otimes V_{1} \otimes V_{2}^{*}$.
  • Figure 3: The quiver $G$ we use in this paper.

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 3.1
  • proof
  • Theorem 4.1: Rephrase of FGS19
  • ...and 22 more