What It Costs to Copy a Hybrid Quantum Head

Fourier analysis places angle-encoded variational heads in known trigonometric spaces; we quantify explicit copying for a three-layer $R_Y$/CNOT head and test the input-design result on a StronglyEntanglingLayers variant. The primary head's enumerated basis reproduces analytic outputs to machine precision for $d=2,\ldots,6$, with $3^d-1$ nonconstant columns. At equal budgets (32{,}400 shots and $3^d$ queries), an orthogonal Fourier grid beats a square random design in all 200 paired seeds at $d=3,4,5$ (geometric-mean NMSE ratio $3.6\times10^{-4}$--$2.7\times10^{-3}$). A second entangler family shows the same preference at $d=3$. At $d=4$, median NMSE is 0.0237 versus 21.0; an 8$\times$-input random control gives 0.0277. The grid removes the heavy-tailed failure by controlling design conditioning. An oracle-assisted tensor-train audit finds no coefficient-storage reduction for the depth-3 head at $d=8$, though a depth-1 head compresses strongly. Random depth-3 heads use 80/81 features at $d=4$, and re-uploaded or non-integer encodings require a different spectrum. These results characterize this surrogate and sampling protocol, not a lower bound on classical copying or quantum advantage.