License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2021.31
URN: urn:nbn:de:0030-drops-135702
URL: https://drops.dagstuhl.de/opus/volltexte/2021/13570/
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Grochow, Joshua A. ; Qiao, Youming

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

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LIPIcs-ITCS-2021-31.pdf (0.6 MB)


Abstract

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).

BibTeX - Entry

@InProceedings{grochow_et_al:LIPIcs.ITCS.2021.31,
  author =	{Joshua A. Grochow and Youming Qiao},
  title =	{{On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{31:1--31:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{James R. Lee},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13570},
  URN =		{urn:nbn:de:0030-drops-135702},
  doi =		{10.4230/LIPIcs.ITCS.2021.31},
  annote =	{Keywords: complexity class, tensor isomorphism, polynomial isomorphism, group isomorphism, stochastic local operations and classical communication}
}

Keywords: complexity class, tensor isomorphism, polynomial isomorphism, group isomorphism, stochastic local operations and classical communication
Collection: 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
Issue Date: 2021
Date of publication: 04.02.2021


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