Separating Quantum and Classical Advice with Good Codes

2026-02-10

AI Summary

This work investigates the fundamental distinctions between quantum and classical proofs—specifically, QMA versus QCMA—and between quantum and classical advice, namely BQP/qpoly versus BQP/poly, in computational complexity theory. By constructing a purely classical oracle based on the Yamakawa–Zhandry code-word intersection problem and leveraging codes with strong list-recovery properties, the study achieves the first unconditional separation of BQP/qpoly from BQP/poly relative to a classical oracle, while also establishing a strict separation between QMA and QCMA. This approach improves upon prior techniques that relied on quantum or restricted classical oracles, offering a more streamlined and general framework that significantly advances our understanding of the power and limitations of quantum versus classical auxiliary computational models.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

We show an unconditional classical oracle separation between the class of languages that can be verified using a quantum proof ($\mathsf{QMA}$) and the class of languages that can be verified with a classical proof ($\mathsf{QCMA}$). Compared to the recent work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), our proof is conceptually and technically simpler, and readily extends to other oracle separations. In particular, our techniques yield the first unconditional classical oracle separation between the class of languages that can be decided with quantum advice ($\mathsf{BQP}/\mathsf{qpoly}$) and the class of languages that can be decided with classical advice ($\mathsf{BQP}/\mathsf{poly}$), improving on the quantum oracle separation of Aaronson and Kuperberg (CCC 2007) and the classically-accessible classical oracle separation of Li, Liu, Pelecanos and Yamakawa (ITCS 2024). Our oracles are based on the code intersection problem introduced by Yamakawa and Zhandry (FOCS 2022), combined with codes that have extremely good list-recovery properties.

Problem

• QMA
• QCMA
• BQP/qpoly
• oracle separation
• quantum advice

Innovation

• oracle separation
• quantum advice
• classical advice
• list-recovery codes
• code intersection problem