A review paper analyzing quantum neural networks (QNNs) through the lens of Fourier analysis. Parameterized quantum circuits (PQCs) produce outputs expressible as finite Fourier series, where the encoding Hamiltonian determines accessible frequencies and training estimates Fourier coefficients. This perspective explains how QNNs can overcome spectral bias that limits classical deep neural networks via 'frequency pinching' — dynamically restricting the model's frequency set to task-relevant bands. Applications explored include quantum ODE solvers (QODE), quantum convolutional neural networks (QCNNs), and quantum reinforcement learning (QRL). The paper also analyzes the barren plateau problem through both Lie-algebraic and Fourier lenses, showing it is fundamentally a spectral collapse, and surveys mitigation strategies including local cost functions, careful initialization, and problem-specific architecture design. Generalization behavior is analyzed via benign overfitting theory and approximation bounds.
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