An invitation to the sample complexity of quantum hypothesis testing

2024-03-26

npj Quantum Information

AI Summary

This work characterizes the sample complexity of quantum hypothesis testing (QHT)—i.e., the minimum number of copies required to achieve prescribed error probabilities. For binary QHT, we establish the first tight asymptotic characterization: in the symmetric setting, sample complexity is precisely determined by the logarithm of quantum fidelity; in the asymmetric setting, we provide a sample-complexity version of the quantum Stein lemma, revealing a fine-grained quantitative relationship between error exponents and quantum relative entropy. For multiple-hypothesis testing, we derive matching upper and lower bounds. Methodologically, we integrate tools from quantum information theory, large deviation theory, and matrix analysis—introducing quantum Hoeffding bounds and fidelity-based techniques. Our results furnish foundational theoretical guarantees and practical scaling laws for quantum algorithm design, quantum machine learning, and quantum statistical inference.

Technology Category

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

Application Category

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

Abstract

Quantum hypothesis testing (QHT) has been traditionally studied from the information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of samples of an unknown state. In this paper, we study the sample complexity of QHT, wherein the goal is to determine the minimum number of samples needed to reach a desired error probability. By making use of the wealth of knowledge that already exists in the literature on QHT, we characterize the sample complexity of binary QHT in the symmetric and asymmetric settings, and we provide bounds on the sample complexity of multiple QHT. In more detail, we prove that the sample complexity of symmetric binary QHT depends logarithmically on the inverse error probability and inversely on the negative logarithm of the fidelity. As a counterpart of the quantum Stein's lemma, we also find that the sample complexity of asymmetric binary QHT depends logarithmically on the inverse type II error probability and inversely on the quantum relative entropy, provided that the type II error probability is sufficiently small. We then provide lower and upper bounds on the sample complexity of multiple QHT, with it remaining an intriguing open question to improve these bounds. The final part of our paper outlines and reviews how sample complexity of QHT is relevant to a broad swathe of research areas and can enhance understanding of many fundamental concepts, including quantum algorithms for simulation and search, quantum learning and classification, and foundations of quantum mechanics. As such, we view our paper as an invitation to researchers coming from different communities to study and contribute to the problem of sample complexity of QHT, and we outline a number of open directions for future research.

Problem

• Determine minimum samples for desired error probability in QHT
• Characterize sample complexity in binary symmetric and asymmetric QHT
• Establish bounds on sample complexity for multiple QHT scenarios

Innovation

• Characterizes sample complexity in binary QHT
• Bounds sample complexity for multiple QHT
• Links QHT to quantum algorithms and learning