Gradients and frequency profiles of quantum re-uploading models

TL;DR

This paper investigates the trainability and expressivity of quantum re-uploading models, providing theoretical bounds and insights into their limitations and robustness, supported by numerical experiments.

Contribution

It introduces the concept of absorption witness for gradient differences and proves that re-uploading models have limited high-frequency expressivity, enhancing understanding of their capabilities.

Findings

Gradient bounds between data-less and re-uploading models

Re-uploading models have functions with limited high-frequency components

Numerical experiments support theoretical bounds

Abstract

Quantum re-uploading models have been extensively investigated as a form of machine learning within the context of variational quantum algorithms. Their trainability and expressivity are not yet fully understood and are critical to their performance. In this work, we address trainability through the lens of the magnitude of the gradients of the cost function. We prove bounds for the differences between gradients of the better-studied data-less parameterized quantum circuits and re-uploading models. We coin the concept of {\sl absorption witness} to quantify such difference. For the expressivity, we prove that quantum re-uploading models output functions with vanishing high-frequency components and upper-bounded derivatives with respect to data. As a consequence, such functions present limited sensitivity to fine details, which protects against overfitting. We performed numerical experiments extending the theoretical results to more relaxed and realistic conditions. Overall, future designs of quantum re-uploading models will benefit from the strengthened knowledge delivered by the uncovering of absorption witnesses and vanishing high frequencies.