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 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.
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BibTeX
@misc{asli2026hierarchical,
title = {{Hierarchical Fourier Approximation for Variational Quantum Distribution Learning}},
author = {Taha Hoseinpour Asli and Sajjad Hashemian and Ebrahim Ardeshir-Larijani},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06307},
eprint = {2609.06307},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06307},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hierarchical Fourier Approximation for Variational Quantum Distribution Learning
AU - Taha Hoseinpour Asli
AU - Sajjad Hashemian
AU - Ebrahim Ardeshir-Larijani
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06307
UR - https://arxiv.org/abs/2609.06307
AB - 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 ℓ_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.
ER -