is still not well-understood, as establishing gap-preserving reductions in the quantum setting presents new challenges. In this paper, we introduce a framework to study such reductions and use it to establish MIP

^*

-completeness of the gapped promise problem for the natural class of independent set games. In such a game, the goal is to determine whether a given graph contains an independent set of a specified size. We construct families of independent set games with constant question size for which the gapped promise problem is undecidable. In contrast, the same problem is decidable in polynomial time in the classical setting. To carry out our reduction, we establish a new stability theorem, which could be of independent interest, allowing us to perturb families of almost PVMs to genuine PVMs.

DOI: 10.48550/arxiv.2505.05253

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Perfect Zero Knowledge for Quantum Multiprover Interactive Proofs

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arXiv:1905.11280Annual Symposium on Foundations of Computer ScienceNov 1, 201925 citations

On the Complexity of Zero Gap MIP*

Hamoon Mousavi, Seyed Sajjad Nezhadi, Henry Yuen

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 $Π_2^0$ , 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.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 $Π_2^0$, 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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The status of the quantum PCP conjecture (games version)

Anand Natarajan, Chinmay Nirkhe

In classical complexity theory, the two definitions of probabilistically checkable proofs -- the constraint satisfaction and the nonlocal games version -- are computationally equal in power. In the quantum setting, the situation is far less clear. The result MIP* = RE of Ji et. al. (arXiv:2001.04383) and refinements by Natarajan and Zhang (arXiv:2302.04322) show that multiprover interactive proof systems with polylogarithmically long messages can solve any decision problem in RE, including undecidable problems like the halting problem. These results show that any connection between the "constraint satisfaction" or "Hamiltonian" quantum PCP conjecture and nonlocal games must involve restricting the players in the game to be computationally efficient. This note contains two main results: (1) we give a "quantum games PCP for AM" in the form of a new construction of a succinct MIP* protocol with efficient provers for the canonical AM-complete problem, and (2) we explain an error in the energy amplification procedure of Natarajan and Vidick (arXiv:1710.03062) which invalidates their claim to have constructed a quantum games PCP for a QMA-complete problem. In surveying the obstacles remaining towards a quantum games PCP for QMA, we highlight the importance and challenge of understanding gap amplification for Hamiltonians even when locality is replaced by much weaker constraints, such as bounds on the "Pauli spectrum" of the Hamiltonian. We hope these questions will motivate progress towards new "baby versions" of Hamiltonian quantum PCP conjecture.In classical complexity theory, the two definitions of probabilistically checkable proofs -- the constraint satisfaction and the nonlocal games version -- are computationally equal in power. In the quantum setting, the situation is far less clear. The result MIP* = RE of Ji et. al. (arXiv:2001.04383) and refinements by Natarajan and Zhang (arXiv:2302.04322) show that multiprover interactive proof systems with polylogarithmically long messages can solve any decision problem in RE, including undecidable problems like the halting problem. These results show that any connection between the "constraint satisfaction" or "Hamiltonian" quantum PCP conjecture and nonlocal games must involve restricting the players in the game to be computationally efficient. This note contains two main results: (1) we give a "quantum games PCP for AM" in the form of a new construction of a succinct MIP* protocol with efficient provers for the canonical AM-complete problem, and (2) we explain an error in the energy amplification procedure of Natarajan and Vidick (arXiv:1710.03062) which invalidates their claim to have constructed a quantum games PCP for a QMA-complete problem. In surveying the obstacles remaining towards a quantum games PCP for QMA, we highlight the importance and challenge of understanding gap amplification for Hamiltonians even when locality is replaced by much weaker constraints, such as bounds on the "Pauli spectrum" of the Hamiltonian. We hope these questions will motivate progress towards new "baby versions" of Hamiltonian quantum PCP conjecture.

arXiv:2403.13084Mar 19, 20241 citation

Compression of quantum multi-prover interactive proofs

Zhengfeng Ji

We present a protocol that transforms any quantum multi-prover interactive proof into a nonlocal game in which questions consist of logarithmic number of bits and answers of constant number of bits. As a corollary, it follows that the promise problem corresponding to the approximation of the nonlocal value to inverse polynomial accuracy is complete for QMIP*, and therefore NEXP-hard. This establishes that nonlocal games are provably harder than classical games without any complexity theory assumptions. Our result also indicates that gap amplification for nonlocal games may be impossible in general and provides a negative evidence for the feasibility of the gap amplification approach to the multi-prover variant of the quantum PCP conjecture.We present a protocol that transforms any quantum multi-prover interactive proof into a nonlocal game in which questions consist of logarithmic number of bits and answers of constant number of bits. As a corollary, it follows that the promise problem corresponding to the approximation of the nonlocal value to inverse polynomial accuracy is complete for QMIP*, and therefore NEXP-hard. This establishes that nonlocal games are provably harder than classical games without any complexity theory assumptions. Our result also indicates that gap amplification for nonlocal games may be impossible in general and provides a negative evidence for the feasibility of the gap amplification approach to the multi-prover variant of the quantum PCP conjecture.

arXiv:1610.03133ACM Symposium on Theory of Computing (STOC)Jun 15, 201727 citations

^*

=RE \cite{ji2020mipre} shows that these are in general undecidable. However, the relative complexity of problems within MIP

^*

^*

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

Π_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.