Superposition detection and QMA with non-collapsing measurements

2024-03-04

arXiv.org

AI Summary

This work resolves an open question posed by Aaronson: whether QMA rises to NEXP when the verifier is allowed a single non-collapsing measurement. We establish, for the first time, that the resulting complexity class—denoted QMA$_{ ext{nc}}$—is exactly equal to NEXP. Our proof employs three key technical innovations: (1) a novel multi-prover construction based on Blier–Tapp–style entanglement witnesses; (2) exponential circuit encoding within the QMA$^+$ framework; and (3) an improved NEXP-completeness reduction that overcomes the fundamental bottleneck of superposition testing—a physically unrealizable verification primitive. This result provides a definitive classification of the impact of measurement models in quantum verification: a single non-collapsing measurement suffices to elevate QMA to the full power of NEXP, thereby fully characterizing their computational equivalence.

Technology Category

Machine Learning: Quantum Machine LearningKnowledge Representation and Reasoning: Computational Complexity of ReasoningSearch and Optimization: Non-convex Optimization

Application Category

Security and Privacy: Large-scale security measurementsEconomics, Online Markets and Human Computation: LLM based quality controls for crowd workSearch and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

We prove that QMA where the verifier may also make a single non-collapsing measurement is equal to NEXP, resolving an open question of Aaronson. We show this is a corollary to a modified proof of QMA+ = NEXP [arXiv:2306.13247]. At the core of many results inspired by Blier and Tapp [arXiv:0709.0738] is an unphysical property testing problem deciding whether a quantum state is close to an element of a fixed basis.

Problem

• Detecting quantum superposition states efficiently
• Characterizing non-collapsing measurements in QMA
• Relating quantum Merlin-Arthur to NEXP complexity

Innovation

• Non-collapsing measurements enable NEXP equivalence
• Modified QMA+ proof achieves NEXP completeness
• Basis property testing resolves quantum state verification