In complexity theory, gap-preserving reductions play a crucial role in studying hardness of approximation and in analyzing the relative complexity of multiprover interactive proof systems. In the quantum setting, multiprover interactive proof systems with entangled provers correspond to gapped promise problems for nonlocal games, and the recent result MIP*=RE [1] shows that these are in general undecidable. However, the relative complexity of problems within MIP* 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 aThis is a striking phenomenlmost PVMs to genuine PVMs.
In complexity theory, gap-preserving reductions play a crucial role in studying hardness of approximation and in analyzing the relative complexity of multiprover interactive proof systems. In the quantum setting, multiprover interactive proof systems with entangled provers correspond to gapped promise problems for nonlocal games, and the recent result MIP*=RE [1] shows that these are in general undecidable. However, the relative complexity of problems within MIP* 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 aThis is a striking phenomenlmost PVMs to genuine PVMs.
In complexity theory, gap-preserving reductions play a crucial role in studying hardness of approximation and in analyzing the relative complexity of multiprover interactive proof systems. In the quantum setting, multiprover interactive proof systems with entangled provers correspond to gapped promise problems for nonlocal games, and the recent result MIP∗=RE \cite{ji2020mipre} shows that these are in general undecidable. However, the relative complexity of problems within MIP∗ 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.
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.
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∃zR(x,y,z). Combined with the previously known result that MIP0co (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.
In this work we consider the interplay between multiprover interactive proofs, quantum entanglement, and zero knowledge proofs - notions that are central pillars of complexity theory, quantum information and cryptography. In particular, we study the relationship between the complexity class MIP*, the set of languages decidable by multiprover interactive proofs with quantumly entangled provers, and the class PZK-MIP*, which is the set of languages decidable by MIP* protocols that furthermore possess the perfect zero knowledge property. Our main result is that the two classes are equal, i.e., MIP* = PZK-MIP*. This result provides a quantum analogue of the celebrated result of Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP = PZK-MIP (in other words, all classical multiprover interactive protocols can be made zero knowledge). We prove our result by showing that every MIP* protocol can be efficiently transformed into an equivalent zero knowledge MIP* protocol in a manner that preserves the completeness-soundness gap. Combining our transformation with previous results, we obtain the corollaries that i) all languages that can be solved in non-deterministic double exponential time have zero knowledge MIP* protocols and ii) all co-recursively enumerable languages (which include undecidable problems as well as all decidable problems) have zero knowledge MIP* protocols with vanishing promise gap.
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.
The complexity of free games with two or more classical players was essentially settled by Aaronson, Impagliazzo, and Moshkovitz (CCC’14). In the quantum world, there are two complexity classes that can be considered quantum analogues of classical free games: (1) AM*, the multiprover interactive proof class corresponding to free games with entangled players, and, somewhat less obviously, (2) BellQMA(2), the class of quantum Merlin-Arthur proof systems with two unentangled Merlins, whose proof states are separately measured by Arthur. In this work, we make significant progress towards a tight characterization of both of these classes. (1) We show a BellQMA(2) protocol for 3SAT on n variables, where the total amount of communication is Õ(√n). This answers an open question of Chen and Drucker (2010) and also shows, conditional on ETH, that the algorithm of Brandão, Christandl and Yard (STOC’11) for optimizing over separable states is tight up to logarithmic factors. (2) We show that AM* with nprovers = 2, question length O(1), and answer-length log(n) is equal to RE, i.e. that free entangled games with constant-sized questions are as powerful as general entangled games. (In contrast, Aaronson, Impagliazzo and Moshkovitz show that classical free games are much weaker than general classical games.) We show this using a question “hyper-compression” theorem that iteratively applies the introspection technique of Ji et al. (2020). Our result is a significant improvement over the headline result of Ji et al., whose MIP* protocol for the halting problem has (n)-sized questions and answers. (3) By the same techniques, we obtain a zero-gap AM* protocol for a Π2 complete language with constant-size questions and almost logarithmically (O(logn · log* n)) large answers, improving on the headline result of Mousavi, Nezhadi and Yuen (STOC’22). (4) Using a connection to the nonuniform complexity of the halting problem we show that any MIP* protocol for RE requires Ω(logn) bits of communication. It follows that our results in item 3 are optimal up to an O(log* n) factor, and that the gapless compression theorems of Mousavi, Nezhadi and Yuen are asymptotically optimal. We conjecture that these bounds can be saturated in the gapped case as well.