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Power and limitations of single-qubit native quantum neural networks

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arxiv 2205.07848 v2 pith:YFGPLFXB submitted 2022-05-16 quant-ph cond-mat.dis-nncs.AIcs.LGmath-phmath.MP

classification quant-phcond-mat.dis-nncs.AIcs.LGmath-phmath.MP
keywords qnnsnetworksneuralquantumsingle-qubitfourierlimitationsnative
verification ladder T0 review T1 audit T2 compute T3 formal
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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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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks

    quant-ph 2025-09 reject novelty 5.0 of 10

    QKANs show strong empirical performance on regression, vision, and language tasks, but the claimed exponential parameter reduction is not rigorously established.

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