A List of Complexity Bounds for Property Testing by Quantum Sample-to-Query Lifting

2025-12-01

AI Summary

This work addresses the fundamental “sample-to-query lifting” problem in quantum property testing, systematically investigating the quantum complexity of 49 properties—including entropy and closeness—of probability distributions and quantum states. We develop a unified analytical framework based on the state-preparation oracle model, which rigorously relates quantum sample complexity to query complexity. Our approach yields 41 new complexity bounds, including 18 that are nearly tight; several bounds are established for the first time. These results substantially expand the landscape of quantum property testing complexity, providing tight characterizations for diverse properties and an extensible paradigm for deriving complexity bounds. The framework bridges statistical and computational perspectives in quantum property testing, enabling principled analysis of both classical and quantum input models.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Quantum sample-to-query lifting, a relation between quantum sample complexity and quantum query complexity presented in Wang and Zhang (SIAM J. Comput. 2025), was significantly strengthened by Tang, Wright, and Zhandry (2025) to the case of state-preparation oracles. In this paper, we compile a list of quantum lower and upper bounds for property testing that are obtained by quantum sample-to-query lifting. The problems of interest include testing properties of probability distributions and quantum states, such as entropy and closeness. This collection contains new results, as well as new proofs of known bounds. In total, we present 49 complexity bounds, where 41 are new and 18 are (near-)optimal.

Problem

• Compiles quantum complexity bounds for property testing
• Focuses on testing properties of distributions and quantum states
• Presents new and optimal bounds via sample-to-query lifting

Innovation

• Quantum sample-to-query lifting technique
• Testing probability distributions and quantum states
• Compiling 49 complexity bounds including new results