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arXiv:2608.01365

Expressive Power and Limitations of Multi-photon Quantum Neural Networks

Zeyu Xiao, Weixu Shi, Yizhi Wang, Lingling Lao, Junjie Wu

quant-phSubmitted Aug 2, 2026• Announced Aug 4, 2026• 0 citations
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Abstract

Quantum neural networks (QNNs) have shown promise in leveraging quantum computation for machine learning tasks. Utilizing multiple identical photons as input, multi-photon quantum neural networks (MPQNNs) have the potential to enhance the expressivity through increasing the photon number. However, how precisely the expressivity of an MPQNN is affected by an increase in photon number, and whether it can be infinitely enhanced by increasing the photon number, remains unexplored. In this work, we quantitatively estimate the expressivity of this model by deriving upper bounds on approximation error in two cases. In the case of a fixed observable, there exists a threshold that scales linearly with the mode number. Below the threshold, the expressivity of an MPQNN can be enhanced polynomially by increasing the photon number. Above the threshold, however, increasing the photon number does not affect the expressivity. In the case of a trainable observable, the expressivity can always be enhanced polynomially by increasing the photon number. These findings are then validated by numerical simulations. Our work elucidates the performance enhancement of multi-photon quantum feature in QNNs, as well as its limitations, offering guidance for leveraging multi-photon advantages in quantum machine learning.

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13 pages,3 figures

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Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.
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Machine learning holds fundamental computational significance due to the increasing demand for efficient solutions to complex tasks in data analysis, pattern recognition, and optimization, which are essential for addressing the multifaceted challenges of modern society. As the volume of data proliferates at an unprecedented rate, the need for more powerful machine learning strategies becomes increasingly evident. Quantum neural networks (QNNs) represent an emerging and transformative research field that seeks to harness the unique principles of quantum mechanics to enhance the capabilities of machine learning algorithms. This survey examines various QNN approaches, including fully connected QNNs, quantum convolutional neural networks, equivariant QNNs, quantum Hopfield networks, quantum Boltzmann machines, quantum reservoir computing, and composite networks for quantum reinforcement learning, quantum generative learning, and quantum transfer learning. We summarize the relevant investigations on their performance, including learning accuracy, training time, and resource requirements, etc. Each QNN type has unique strengths and weaknesses, offering diverse solutions for different applications.