# Fundamental Limits to Expressive Capacity of Finitely Sampled Qubit-Based Systems

**Authors:** Fangjun Hu, Gerasimos Angelatos, Saeed A. Khan, Marti Vives, Esin Türeci, Leon Bello, Graham E. Rowlands, Guilhem J. Ribeill, Hakan E. Türeci  
**arXiv:** [2301.00042](https://arxiv.org/abs/2301.00042)  
**Date:** 2022-12-30  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2301.00042v1/

## Key points

- Quantum machine learning (QML) shows promise for data-driven applications
- Qubit-based systems have limited expressive capacity due to quantum sampling noise
- Framework developed to quantify expressive capacity using projective measurements
- Tight bound on expressive capacity and accuracy limit calculated and compared to experiments
- Low-noise eigentasks improve performance in classification tasks, robust to noise and overfitting
- Entanglement enhances learning capacity by reducing noise in eigentasks
- Implications for quantum machine learning and sensing applications
- Understanding limitations of quantum sampling noise can enhance performance and reliability

## Summary

The field of quantum machine learning (QML) has shown great promise in utilizing quantum systems for data-driven applications. However, the expressive capacity of qubit-based systems is limited by the quantum sampling noise incurred during measurement. Previous studies have suggested that noise affects the resolvable capacity of quantum systems; its precise impact on learning remains an open question. In this study, the authors develop a framework for quantifying the expressive capacity of qubit-based systems using finite numbers of projective measurements. They calculate a tight bound on the expressive capacity and corresponding accuracy limit, which they compare to experiments conducted on superconducting quantum processors. Low-noise eigentasks are found to improve performance in tasks such as classification in a way that is robust to noise and overfitting. Additionally, experimental and numerical analyses suggest that entanglement enhances learning capacity by reducing noise in eigentasks. These findings have broad implications for quantum machine learning and sensing applications. By understanding the limitations imposed by quantum sampling noise and identifying effective strategies to mitigate its effects, researchers can enhance the performance and reliability of qubit-based systems for various data-driven tasks.