Sample Complexity of Composite Quantum Hypothesis Testing

2026-01-13

AI Summary

This work investigates the sample complexity of composite binary quantum hypothesis testing in the finite-sample regime, specifically characterizing the minimum number of copies of an unknown quantum state required to distinguish between two possibly infinite uncertainty sets. The analysis is further extended to the setting of differential privacy. By integrating tools from quantum information theory, minimax analysis, and differential privacy, the study establishes the first tight upper and lower bounds that match up to constant factors in the finite-sample setting. These results not only provide a precise characterization of sample complexity for general uncertainty sets but also lay the theoretical foundations for privacy-preserving composite quantum hypothesis testing, thereby addressing a significant gap in the existing literature.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper investigates symmetric composite binary quantum hypothesis testing (QHT), where the goal is to determine which of two uncertainty sets contains an unknown quantum state. While asymptotic error exponents for this problem are well-studied, the finite-sample regime remains poorly understood. We bridge this gap by characterizing the sample complexity -- the minimum number of state copies required to achieve a target error level. Specifically, we derive lower bounds that generalize the sample complexity of simple QHT and introduce new upper bounds for various uncertainty sets, including of both finite and infinite cardinalities. Notably, our upper and lower bounds match up to universal constants, providing a tight characterization of the sample complexity. Finally, we extend our analysis to the differentially private setting, establishing the sample complexity for privacy-preserving composite QHT.

Problem

• composite quantum hypothesis testing
• sample complexity
• quantum state discrimination
• finite-sample regime
• differential privacy

Innovation

• sample complexity
• composite quantum hypothesis testing
• finite-sample analysis
• differential privacy
• quantum state discrimination