Quantum PCPs: on Adaptivity, Multiple Provers and Reductions to Local Hamiltonians

2024-03-07

arXiv.org

AI Summary

This work addresses three central questions in quantum Probabilistically Checkable Proofs (QPCP): the equivalence of adaptive and nonadaptive QPCP, the complexity characterization of multiprover QPCP, and its connection to local Hamiltonian reductions. We unify the modeling of adaptive and multiprover QPCP, design novel quantum interactive proof protocols, and integrate relativization/de-relativization techniques with constant-query local Hamiltonian reductions. Our contributions are: (1) the first proof that nonadaptive QPCP with constant query complexity can simulate adaptive QPCP; (2) a sufficient condition for QPCP[q] ⊆ QCMA; (3) a proof that if QMA(k) admits a PCP characterization, then QMA(2) = QMA; and (4) an explicit oracle separation demonstrating that QPCP inherently requires nonrelativizing techniques. These results establish tight connections between two major open problems—whether QMA(2) = QMA and whether QPCP is QCMA-solvable—advancing the foundational understanding of quantum proof systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Natural Language Processing: Question Answering

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We define a general formulation of quantum PCPs, which captures adaptivity and multiple unentangled provers, and give a detailed construction of the quantum reduction to a local Hamiltonian with a constant promise gap. The reduction turns out to be a versatile subroutine to prove properties of quantum PCPs, allowing us to show: (i) Non-adaptive quantum PCPs can simulate adaptive quantum PCPs when the number of proof queries is constant. In fact, this can even be shown to hold when the non-adaptive quantum PCP picks the proof indices simply uniformly at random from a subset of all possible index combinations, answering an open question by Aharonov, Arad, Landau and Vazirani (STOC '09). (ii) If the $q$-local Hamiltonian problem with constant promise gap can be solved in $mathsf{QCMA}$, then $mathsf{QPCP}[q] subseteq mathsf{QCMA}$ for any $q in O(1)$. (iii) If $mathsf{QMA}(k)$ has a quantum PCP for any $k leq ext{poly}(n)$, then $mathsf{QMA}(2) = mathsf{QMA}$, connecting two of the longest-standing open problems in quantum complexity theory. Moreover, we also show that there exists (quantum) oracles relative to which certain quantum PCP statements are false. Hence, any attempt to prove the quantum PCP conjecture requires, just as was the case for the classical PCP theorem, (quantumly) non-relativizing techniques.

Problem

• General formulation of quantum PCPs capturing adaptivity and multiple provers
• Reduction to local Hamiltonian with constant promise gap
• Connecting quantum PCPs to longstanding open problems in complexity theory

Innovation

• General quantum PCP formulation with adaptivity
• Reduction to local Hamiltonian with gap
• Non-adaptive PCPs simulate adaptive ones