We study variational quantum distribution learning through a hierarchy of Walsh--Fourier approximations on the Boolean cube. At each level, a selected set of target Fourier coefficients defines a spectral truncation, which is projected onto the probability simplex and used as the target of a quantum circuit Born machine. Parameters learned at one level initialize the next through a warm-start map. We prove an end-to-end expected learning guarantee where the approximation term is determined by the omitted Fourier mass, while a normalized unbiased estimator yields an explicit statistical bound for empirical truncations. We then instantiate the abstract discrepancy conditions for total variation distance and relate the resulting distributional error to quantum-state fidelity. The total-variation specialization incurs the explicit factor 2n−1 under our normalized ℓ2 convention and is therefore informative only for sufficiently concentrated Fourier tails. The framework does not establish global trainability or eliminate barren plateaus; rather, it identifies the conditions under which low-to-high spectral training admits a approximation--estimation--optimization analysis.
Figures & tables
Figure 1: The figure tests the implementation of the target hierarchy. The hierarchy level k on the horizontal axis and ∥pδ,K−Tkpδ,K∥2 on the vertical axis. Show the numerical evaluation and the exact curve for several values of δ , with n , K , and α fixed.
Figure 2: (Left) Convergence of the best achieved loss versus iteration for the Naive, ADAM, and Hierarchical methods. The faint lines represent the raw loss, while the solid lines indicate the cumulative minimum loss. (Top Right) Magnitude of the normalized gradient ∥∇new∥ at the expansion point, comparing the Naive and Hierarchical approaches. (Bottom Right) The relationship between the hierarchy level and the trap density, showing how the spectral truncation affects the density of the approximation space via the hierarchy level.
Figure 3: Mean normalized ℓ2 estimation error is shown as a function of the sample size m on logarithmic axes, with n , K , k , α , and δ held fixed. We compare the raw truncation error, ∥Tkp−pk∥2 , with the error after projection onto the probability simplex, ∥Tkp−ΠΔ(pk)∥2 . shaded intervals denote confidence intervals over independent data samples. The m−1/2 reference line highlights the expected Monte Carlo scaling.