Architecture Shape Governs QNN Trainability: Jacobian Null Space Growth and Parameter Efficiency

Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging $N$ qubits with $L$ encoding layers each – sharing encoding budget $E=NL$ – generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape $(N,L)$ at fixed $E$. We identify structural rank deficiency of the coefficient matching Jacobian $J$ as the mechanism responsible. For serial single-qubit architectures, we prove $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(J)\le 2L+1$ regardless of parameter count $P$, with $\dim (\ker J)\ge P-(2L+1)$ growing without bound – a phenomenon we term \emph{structural gradient starvation}: a growing fraction of parameters become structurally decoupled from the loss as $P$ increases at fixed $L$. Parallel architectures avoid this via independent phase trajectories, ensuring ${\sigma }_{\min }(J^{(\mathrm{p}\mathrm{a}\mathrm{r})})>0$ generically for $P\le 2E+1$, so no parameter lies in $\ker J$. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves $R^2\ge 0.95$ with $1.6$–$2.2\times$ fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.