Geometric Origin of Fourier Coefficients in Data-Reuploading Single-Qubit Circuits

In this work, we characterize the Fourier structure of single-qubit, data-reuploading parameterized quantum circuits (PQCs). Each data-encoding rotation constrains the Bloch vector to a circle whose radius is fixed by the trainable blocks. We prove the Bloch Circle Projection Theorem: for a single layer, projecting this trajectory onto the measurement axis directly yields the circuit’s Fourier coefficients. This gives a concrete design rule: choose rotation axes non-collinear with the encoding axis to avoid degenerate projections. The same geometric argument extends to \(L\) -layer circuits; permitting layer-wise flexible rotation axes preserves full expressivity, even for targets with rich high-frequency content.