On the Complexity of Zero Gap MIP*

Authors Hamoon Mousavi, Seyed Sajjad Nezhadi, Henry Yuen

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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


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.

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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)


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.

Subject Classification

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


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