Quantum neural networks employ parametrized quantum circuits (PQCs) to map data inputs to desired predictions. Recent analytical research demonstrates that PQCs consisting of data reuploading structure are naturally expressed as partial Fourier series and that a single-qubit circuit can serve as a universal approximator for univariate functions. However, this prior work largely focuses on representational capacity and does not provide intuitive or structural explanations of how the individual circuit components govern the resulting Fourier coefficients. In this paper, we peel back the black box and analyze the intrinsic structure of PQCs. In particular, we investigate how data encoding, repeated reuploading, and trainable unitary operators combine to represent function classes characterized by a Fourier expansion with specific accessible frequency components. Our key contribution is to show, both mathematically and empirically, that the configuration of trainable parameters creates an output qubit trajectory that is critical for the representation of Fourier coefficients. We further introduce a strategy for designing a data encoder with tunable scaler, which ensures that the circuit generalizes its mapping ability to capture a target frequency spectrum of the input data.

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• Quantum circuits