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.
The derandomization of MA, the probabilistic version of NP, is a long standing open question. In this work, we connect this problem to a variant of another major problem: the quantum PCP conjecture. Our connection goes through the surprising quantum characterization of MA by Bravyi and Terhal. They proved the MA-completeness of the problem of deciding whether the groundenergy of a uniform stoquastic local Hamiltonian is zero or inverse polynomial. We show that the gapped version of this problem, i.e. deciding if a given uniform stoquastic local Hamiltonian is frustration-free or has energy at least some constant ε, is in NP. Thus, if there exists a gap-amplification procedure for uniform stoquastic Local Hamiltonians (in analogy to the gap amplification procedure for constraint satisfaction problems in the original PCP theorem), then MA = NP (and vice versa). Furthermore, if this gap amplification procedure exhibits some additional (natural) properties, then P = RP. We feel this work opens up a rich set of new directions to explore, which might lead to progress on both quantum PCP and derandomization. We also provide two small side results of potential interest. First, we are able to generalize our result by showing that deciding if a uniform stoquastic Local Hamiltonian has negligible or constant frustration can be also solved in NP. Additionally, our work reveals a new MA-complete problem which we call SetCSP, stated in terms of classical constraints on strings of bits, which we define in the appendix. As far as we know this is the first (arguably) natural MA-complete problem stated in non-quantum CSP language.
The quantum PCP (QPCP) conjecture states that all problems in QMA, the quantum analogue of NP, admit quantum verifiers that only act on a constant number of qubits of a polynomial size quantum proof and have a constant gap between completeness and soundness. Despite an impressive body of work trying to prove or disprove the quantum PCP conjecture, it still remains widely open. The above-mentioned proof verification statement has also been shown equivalent to the QMA-completeness of the Local Hamiltonian problem with constant relative gap. Nevertheless, unlike in the classical case, no equivalent formulation in the language of multi-prover games is known. In this work, we propose a new type of quantum proof systems, the Pointer QPCP, where a verifier first accesses a classical proof that he can use as a pointer to which qubits from the quantum part of the proof to access. We define the Pointer QPCP conjecture, that states that all problems in QMA admit quantum verifiers that first access a logarithmic number of bits from the classical part of a polynomial size proof, then act on a constant number of qubits from the quantum part of the proof, and have a constant gap between completeness and soundness. We define a new QMA-complete problem, the Set Local Hamiltonian problem, and a new restricted class of quantum multi-prover games, called CRESP games. We use them to provide two other equivalent statements to the Pointer QPCP conjecture: the Set Local Hamiltonian problem with constant relative gap is QMA-complete; and the approximation of the maximum acceptance probability of CRESP games up to a constant additive factor is as hard as QMA. Our new conjecture is weaker than the original QPCP conjecture and hence provides a natural intermediate step towards proving the quantum PCP theorem. Furthermore, this is the first equivalence between a quantum PCP statement and the inapproximability of quantum multi-prover games.
We show that given an explicit description of a multiplayer game, with a classical verifier and a constant number of players, it is QMA-hard, under randomized reductions, to distinguish between the cases when the players have a strategy using entanglement that succeeds with probability 1 in the game, or when no such strategy succeeds with probability larger than 1/2. This proves the “games quantum PCP conjecture” of Fitzsimons and the second author (ITCS'15), albeit under randomized reductions. The core component in our reduction is a construction of a family of two-player games for testing n-qubit maximally entangled states. For any integer n ≥ 2, we give such a game in which questions from the verifier are O(log n) bits long, and answers are poly(loglogn) bits long. We show that for any constant ε ≥ 0, any strategy that succeeds with probability at least 1 - ε in the test must use a state that is within distance δ(ε) = O(εc) from a state that is locally equivalent to a maximally entangled state on n qubits, for some universal constant c > 0. The construction is based on the classical plane-vs-point test for multivariate low-degree polynomials of Raz and Safra (STOC'97). We extend the classical test to the quantum regime by executing independent copies of the test in the generalized Pauli X and Z bases over Fq, where q is a sufficiently large prime power, and combine the two through a test for the Pauli twisted commutation relations. Our main complexity-theoretic result is obtained by combining this family of games with techniques from the classical PCP literature. More specifically, we use constructions of PCPs of proximity introduced by Ben-Sasson et al. (CCC'05), and crucially rely on a linear property of such PCPs. Another consequence of our results is a deterministic reduction from the games quantum PCP conjecture to a suitable formulation of the constraint satisfaction quantum PCP conjecture.
The study of multiprover interactive proof systems, of locally testable codes, and of property testing are deeply linked, conceptually if not formally, through their role in the proof of the PCP theorem in complexity theory. Recently there has been substantial progress on an analogous research programme in quantum complexity theory. Two years ago we characterized the power of multiprover interactive proof systems with provers sharing entanglement, showing that MIP^* = RE [Ji et al., 2020], a hugely surprising increase in power from the classical result MIP = NEXP of [Babai et al., 1991]. The following year Panteleev and Kalachev gave the first construction of quantum low-density parity-check codes (QLDPC) [Panteleev and Kalachev, 2022], thus marking a major step towards the possible realization of good quantum locally testable codes - the classical analogue of which was only constructed quite recently [Dinur et al., 2022]. And finally, less than a year ago Anshu, Breuckmann and Nirkhe used facts evidenced in the construction of good decoders for the new QLDPC codes to resolve the NLTS conjecture [Anshu et al., 2022], widely viewed as a crucial step on the way to a possible quantum PCP theorem. In the talk I will survey these results, making an effort to motivate and present them to the non-expert. I will explain the connections between them and point to where, in my opinion, our understanding is currently lacking. Along the way I will highlight a number of open problems whose resolution could lead to further progress on one of the most important research programmes in quantum complexity theory.
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.
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.