Compositional Spectral Representations for Efficient Quantum Neural Approximation

Universal approximation is now established for several quantum neural-network families, yet it does not determine the circuit cost of representing a structured target. Constructive spectral models compile global Fourier or Chebyshev expansions; products and compositions of simple local factors can make these expansions exponentially long while the generating computation remains small. We introduce stable quantum spectral formula networks (QSFN), which compile the bounded generating formula directly, making circuit resources follow formula complexity even when its global spectrum is exponentially long. Univariate spectral leaves expose signed Pauli-Z expectations, and internal vertices combine them through bounded maps that are affine in each input separately. We prove that these maps are exactly the transformations compatible with the interface, establish universal and quantitative analytic approximation, and give a deterministic circuit compiler with explicit query, gate, depth, normalization, and stability guarantees. For an explicit continuous bounded family, every fixed-error signed-flat Fourier approximation requires $\Omega (2^d)$ branches. QSFN gives an exact, locally query-optimal $\Theta (d^2)$ construction with unit normalization. Across three experiments, matched-error comparisons reduce entangling depth by up to 10.5-fold, exact semantic controls isolate factorization as the source of compression, and fixed-error tests reproduce the predicted polynomial-versus-exponential scaling. The results establish compositional structure as an independent determinant of quantum approximation cost.