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From Holant to Quantum Entanglement and Back

Authors Jin-Yi Cai, Zhiguo Fu, Shuai Shao

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Holant problem
  • Quantum entanglement
  • SLOCC equivalence
  • Bell property

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Abstract

Holant problems are intimately connected with quantum theory as tensor networks. We first use techniques from Holant theory to derive new and improved results for quantum entanglement theory. We discover two particular entangled states |Ψ₆⟩ of 6 qubits and |Ψ₈⟩ of 8 qubits respectively, that have extraordinary closure properties in terms of the Bell property. Then we use entanglement properties of constraint functions to derive a new complexity dichotomy for all real-valued Holant problems containing a signature of odd arity. The signatures need not be symmetric, and no auxiliary signatures are assumed.

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Jin-Yi Cai, Zhiguo Fu, and Shuai Shao. From Holant to Quantum Entanglement and Back. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.22

Author Details

Jin-Yi Cai
  • Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI, USA
Zhiguo Fu
  • School of Information Science and Technology and KLAS, Northeast Normal University, Changchun, China
Shuai Shao
  • Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI, USA

Funding

  • Cai, Jin-Yi: Jin-Yi Cai is supported by NSF CCF-1714275.
  • Fu, Zhiguo: Zhiguo Fu is supported by NSFC-61872076.
  • Shao, Shuai: Shuai Shao is supported by NSF CCF-1714275.

Acknowledgements

We want to thank anonymous reviewers for their helpful comments.

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