Regularity determines the quantum loading cost of rough functions

We determine the gate complexity of amplitude encoding (loading) for rough functions. For a function of Hölder regularity s > 0 and target infidelity δ, the worst-case cost is Θ̃(δ^{-1/(2s)}), independent of the number of qubits n for every s > 0. The tilde hides factors polylogarithmic in n and 1/δ. Upper bound. The construction is explicit. The function is approximated by piecewise polynomials of degree below ⌈s⌉ on 2^J dyadic intervals. The resulting state is an arbitrary state on about J qubits, followed by a fixed polynomial isometry that is the same for every interval and acts on only O(log(1/δ)) qubits; the remaining qubits are put in |+⟩ by Hadamard gates. No periodicity and no quantum wavelet transform are needed. For 0 < s ≤ 1 the construction reduces to Haar coarse-graining. Lower bound. A packing argument over sign patterns gives the matching lower bound. It holds for any finite gate set and allows ancillas and mixed outputs. A Gaussian version, based on Anderson’s inequality and small-ball probabilities, extends it to correlated Gaussian vectors. Combined with a bound on the smallest eigenvalue of the level-wise Haar covariance, it shows that loading a given fractional Brownian path with Hurst index H costs Θ̃(δ^{-1/(2H)}) with high probability. For rough-volatility paths (H ≈ 0.1) this is δ^{-5}