Hierarchical Fourier Approximation for Variational Quantum Distribution Learning

• URL: http://arxiv.org/abs/2609.06307v1
• Date: Sat, 05 Sep 2026 23:34:49 GMT
• Title: Hierarchical Fourier Approximation for Variational Quantum Distribution Learning
• Abstract summary: We study quantum distribution learning through a hierarchy of Walsh-Fourier approximations on the Boolean cube.<n>We prove an end-to-end expected learning guarantee where the approximation term is determined by the omitted Fourier mass.<n>We identify the conditions under which low-to-high spectral training admits approximation-estimation-optimization analysis.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $2^{n-1}$ under our normalized $\ell_2$ 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.

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