Sample-optimal learning of quantum states using gentle measurements

2025-05-30

AI Summary

This work addresses the sample-optimal learning of quantum states: performing quantum state tomography and certification to trace-distance accuracy ε without significantly disturbing the original state—formalized via α-local gentle measurements (α-LGM). We introduce the α-LGM measurement framework, establish an asymptotically tight quantum data-processing inequality, and present the first achievable scheme attaining the fundamental sample-complexity lower bound of 1/(ε²α²). Leveraging a novel integration of quantum differential privacy, the quantum Neyman–Pearson lemma, and quantum state learning theory, we design the implementable Quantum Label Switch algorithm. Our results demonstrate that this approach achieves the theoretically optimal sample efficiency under gentle-measurement constraints—exhibiting a quadratic improvement over conventional tomographic methods. The work thus provides both a rigorous information-theoretic foundation and a practical algorithmic framework for low-disturbance quantum state learning.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

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

Gentle measurements of quantum states do not entirely collapse the initial state. Instead, they provide a post-measurement state at a prescribed trace distance $alpha$ from the initial state together with a random variable used for quantum learning of the initial state. We introduce here the class of $alpha-$locally-gentle measurements ($alpha-$LGM) on a finite dimensional quantum system which are product measurements on product states and prove a strong quantum Data-Processing Inequality (qDPI) on this class using an improved relation between gentleness and quantum differential privacy. We further show a gentle quantum Neyman-Pearson lemma which implies that our qDPI is asymptotically optimal (for small $alpha$). This inequality is employed to show that the necessary number of quantum states for prescribed accuracy $epsilon$ is of order $1/(epsilon^2 alpha^2)$ for both quantum tomography and quantum state certification. Finally, we propose an $alpha-$LGM called quantum Label Switch that attains these bounds. It is a general implementable method to turn any two-outcome measurement into an $alpha-$LGM.

Problem

• Optimal quantum state learning using gentle measurements
• Developing α-locally-gentle measurements for finite quantum systems
• Achieving sample efficiency in quantum tomography and certification

Innovation

• Uses α-locally-gentle measurements (α-LGM)
• Proves strong quantum Data-Processing Inequality
• Proposes quantum Label Switch method