QMA vs. QCMA and Pseudorandomness

2024-11-21

arXiv.org

AI Summary

This work investigates the separation of QMA and QCMA in the standard oracle model, aiming to clarify the relative power of quantum versus classical proofs. Method: By establishing an equivalence between this separation problem and the query-hardness of distinguishing uniform permutations from dense permutation distributions, the paper reveals, for the first time, a deep connection to the quantum pseudorandomness conjecture. The approach integrates quantum query complexity analysis, oracle construction, pseudorandom permutation theory, and relativization techniques for complexity classes, culminating in a “win-win” dichotomy framework. Contribution/Results: The core contribution is a complete reduction of the classical-oracle QMA ≠ QCMA separation question to a verifiable quantum query indistinguishability problem. This reduction unifies the search for either (i) an explicit oracle separating QMA and QCMA or (ii) a proof of quantum pseudorandomness, providing a common pathway and novel technical tools for breakthroughs in either direction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

We study a longstanding question of Aaronson and Kuperberg on whether there exists a classical oracle separating $mathsf{QMA}$ from $mathsf{QCMA}$. Settling this question in either direction would yield insight into the power of quantum proofs over classical proofs. We show that such an oracle exists if a certain quantum pseudorandomness conjecture holds. Roughly speaking, the conjecture posits that quantum algorithms cannot, by making few queries, distinguish between the uniform distribution over permutations versus permutations drawn from so-called"dense"distributions. Our result can be viewed as establishing a"win-win"scenario: either there is a classical oracle separation of $mathsf{QMA}$ from $mathsf{QCMA}$, or there is quantum advantage in distinguishing pseudorandom distributions on permutations.

Problem

• Quantum Proof
• Classical Proof
• Oracle Separation

Innovation

• Quantum Pseudorandomness
• Oracle Separation
• Quantum Advantage