Quantum models pay for non-linearity in data encodings

Ancillas, mid-circuit measurements, noise, post-selection and classical post-processing are each credited with making quantum machine-learning models non-linear in their input. Here we show that, when the input enters only through Pauli encoding gates, these all draw on one budget: the spectral range of the encodings in each shot. For any procedure with finitely many shots, arbitrary channels, feed-forward and a bounded classical head, the expected output extends to complex inputs a+ib as an entire function bounded by ∏ᵢ cosh(Eᵢb), where Eᵢ is the range of shot i, and its Fourier coefficients lie below a Gaussian whose variance is the sum of the Eᵢ². Range spent inside one shot buys sharp features at the Heisenberg rate, range spread over shots only at the shot-noise rate, so a hybrid model that encodes once per shot needs quadratically more encodings than a coherent circuit to resolve a jump. What range costs depends on the regime: circuit depth, capped by coherence, when the encoding scale is fixed, and input precision when it can be rescaled. Post-selection escapes the budget only with infinitely many shots. Sixteen experiments on IBM, IQM and IonQ processors, fifteen of them pre-registered, test the predicted rates and costs.