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.ICALP.2020.87
URN: urn:nbn:de:0030-drops-124940
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12494/
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Mousavi, Hamoon ; Nezhadi, Seyed Sajjad ; Yuen, Henry

On the Complexity of Zero Gap MIP*

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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.

BibTeX - Entry

@InProceedings{mousavi_et_al:LIPIcs:2020:12494,
  author =	{Hamoon Mousavi and Seyed Sajjad Nezhadi and Henry Yuen},
  title =	{{On the Complexity of Zero Gap MIP*}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{87:1--87:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12494},
  URN =		{urn:nbn:de:0030-drops-124940},
  doi =		{10.4230/LIPIcs.ICALP.2020.87},
  annote =	{Keywords: Quantum Complexity, Multiprover Interactive Proofs, Computability Theory}
}

Keywords: Quantum Complexity, Multiprover Interactive Proofs, Computability Theory
Collection: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue Date: 2020
Date of publication: 29.06.2020

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