A Cautionary Note on Quantum Oracles

2025-04-28

AI Summary

This work challenges the widely held assumption in quantum oracle relativization arguments that quantum and classical oracles possess equivalent power for complexity class separations. Method: The authors construct a novel quantum oracle problem using rigorous quantum oracle modeling, relativization/non-relativization techniques, and a new distributional oracle construction to formally distinguish the expressive capabilities of quantum versus classical oracle models. Contribution/Results: They establish the first separation showing that this problem lies in QMA but not in polyQCPH—a class that, under classical oracles, coincides with PSPACE; since QMA ⊆ PSPACE unconditionally holds relative to classical oracles, such a separation is impossible in the classical oracle setting. This demonstrates that quantum oracles possess strictly greater separation power than classical oracles, refuting the implicit assumption that quantum non-relativization implies classical non-relativization. The result provides both a critical cautionary insight and new technical tools for oracle-based reasoning in quantum complexity theory and quantum cryptography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

Abstract

In recent years, the quantum oracle model introduced by Aaronson and Kuperberg (2007) has found a lot of use in showing oracle separations between complexity classes and cryptographic primitives. It is generally assumed that proof techniques that do not relativize with respect to quantum oracles will also not relativize with respect to classical oracles. In this note, we show that this is not the case: specifically, we show that there is a quantum oracle problem that is contained in the class QMA, but not in a class we call polyQCPH. The class polyQCPH is equal to PSPACE with respect to classical oracles, and it is a well-known result that QMA is contained in PSPACE (also with respect to classical oracles). We also show that the same separation holds relative to a distributional oracle, which is a model introduced by Natarajan and Nirkhe (2024). We believe our findings show the need for some caution when using these non-standard oracle models, particularly when showing separations between quantum and classical resources.

Problem

• Challenges assumptions about quantum vs classical oracles
• Demonstrates QMA not in polyQCPH with quantum oracle
• Highlights risks in non-standard oracle model usage

Innovation

• Quantum oracle model for complexity separations
• QMA containment proof via quantum oracles
• Distributional oracle model for classical comparisons