Pointer Quantum PCPs and Multi-Prover Games

Abstract

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. This is the first equivalence between a quantum PCP statement and the inapproximability of quantum multi-prover games.