Hierarchical Fourier Approximation for Variational Quantum Distribution Learning

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.

Quantum Physics (quant-ph); Machine Learning (cs.LG)

arXiv:2609.06307 [quant-ph]

(or arXiv:2609.06307v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.06307 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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