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On the Complexity of Zero Gap MIP*

Authors Hamoon Mousavi, Seyed Sajjad Nezhadi, Henry Yuen

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ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Theory of computation → Computability
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum Complexity
  • Multiprover Interactive Proofs
  • Computability Theory

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Abstract

The class MIP^* is the set of languages decidable by multiprover interactive proofs with quantum entangled provers. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that MIP^* is equal to RE, the set of recursively enumerable languages. In particular this shows that the complexity of approximating the quantum value of a non-local game G is equivalent to the complexity of the Halting problem. 
In this paper we investigate the complexity of deciding whether the quantum value of a non-local game G is exactly 1. This problem corresponds to a complexity class that we call zero gap MIP^*, denoted by MIP₀^*, where there is no promise gap between the verifier’s acceptance probabilities in the YES and NO cases. We prove that MIP₀^* extends beyond the first level of the arithmetical hierarchy (which includes RE and its complement coRE), and in fact is equal to Π₂⁰, the class of languages that can be decided by quantified formulas of the form ∀ y ∃ z R(x,y,z).
Combined with the previously known result that MIP₀^{co} (the commuting operator variant of MIP₀^*) is equal to coRE, our result further highlights the fascinating connection between various models of quantum multiprover interactive proofs and different classes in computability theory.

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Hamoon Mousavi, Seyed Sajjad Nezhadi, and Henry Yuen. On the Complexity of Zero Gap MIP*. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 87:1-87:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.87

Author Details

Hamoon Mousavi
  • Department of Computer Science, University of Toronto, Canada
Seyed Sajjad Nezhadi
  • Department of Computer Science, University of Toronto, Canada
Henry Yuen
  • Department of Computer Science and Department of Mathematics, University of Toronto, Canada

Funding

  • Mousavi, Hamoon: was supported by the Ontario Graduate Scholarship (OGS).
  • Yuen, Henry: was supported by NSERC Discovery Grant 2019-06636.

Acknowledgements

We thank Matt Coudron, Thomas Vidick, and especially William Slofstra for numerous helpful discussions. We also thank the reviewers of ICALP 2020 for suggestions to improve the presentation.

References

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