Gradients and frequency profiles of quantum re-uploading models
1$⟨aQa^L⟩$ Applied Quantum Algorithms, Universiteit Leiden
2Quantum Technology Initiative, CERN, Geneva, Switzerland
3Instituut-Lorentz, Universiteit Leiden, the Netherlands
| Published: | 2024-11-14, volume 8, page 1523 |
| Eprint: | arXiv:2311.10822v2 |
| Doi: | https://doi.org/10.22331/q-2024-11-14-1523 |
| Citation: | Quantum 8, 1523 (2024). |
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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 $\textit{absorption witness}$ 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.
Featured image: LEFT: Re-uploading models are generated by adding data to a fixed parameterized quantum circuit. Trainability properties are inherited if the data can be re-absorbed as a parameter shift such as in PQC2.
RIGHT: The Fourier transform of quantum re-uploading models exhibits Gaussian profiles if the data uploading gates are interleaved with random trainable gates.
Popular summary
In terms of trainability, we demonstrate how the presence of data affects the phenomenon of vanishing gradients in comparison to scenarios without data. We introduce the concept of "absorption witness" which quantifies this difference when averaging over parameters. On the expressivity front, we derive the average Fourier spectrum of the hypothesis function from the circuit architecture. This profile rapidly tends towards a Gaussian shape in deep quantum computations. As a consequence, high-frequency terms vanish. This observation hints towards increased generalization capabilities in machine learning tasks.
Our work contributes to the broader understanding of quantum re-uploading models in machine learning, extending the existing knowledge derived from variational quantum algorithms to scenarios involving the presence of data.
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