On the complexity of zero gap MIP*

• URL: http://arxiv.org/abs/2002.10490v2
• Date: Wed, 29 Apr 2020 00:08:38 GMT
• Title: On the complexity of zero gap MIP*
• Authors: Hamoon Mousavi, Seyed Sajjad Nezhadi, and Henry Yuen
• Abstract summary: We show that the class $mathsfMIP*$ is equal to $mathsfRE$. 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.
• Score: 0.11470070927586014
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The class $\mathsf{MIP}^*$ 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 $\mathsf{MIP}^*$ is equal to $\mathsf{RE}$, 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 $\mathsf{MIP}^*$, denoted by $\mathsf{MIP}^*_0$, where there is no promise gap between the verifier's acceptance probabilities in the YES and NO cases. We prove that $\mathsf{MIP}^*_0$ extends beyond the first level of the arithmetical hierarchy (which includes $\mathsf{RE}$ and its complement $\mathsf{coRE}$), and in fact is equal to $\Pi_2^0$, the class of languages that can be decided by quantified formulas of the form $\forall y \, \exists z \, R(x,y,z)$. Combined with the previously known result that $\mathsf{MIP}^{co}_0$ (the commuting operator variant of $\mathsf{MIP}^*_0$) is equal to $\mathsf{coRE}$, 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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