Harmonic analysis and circuit approximation of Sobolev functions on quantum input spaces

Philipp Schmocker

quant-phOct 9, 2026

Harmonic analysis and circuit approximation of Sobolev functions on quantum input spaces

Authors: Philipp Schmocker

Abstract

We derive approximation rates for Sobolev functions defined on the complex unit sphere and complex projective space, representing quantum states, and on the special and projective unitary groups, representing quantum unitary operations. By using harmonic analysis on compact groups and compact homogeneous spaces, we characterize the corresponding Peter--Weyl decompositions and Sobolev spaces via irreducible representations and the Laplace--Beltrami operator. Combining spectral truncation with finite-depth quantum-circuit approximation, invoking the Solovay--Kitaev theorem, we establish quantitative rates that separate the spectral truncation error from the circuit approximation error.

Figures & tables

Figure 1. Empirical $\operatorname{RMSE}_{K,L}$ ( 5.5 ) between $f$ and its approximation $\mathcal{Q}_{K,L}f$ depending on the maximal circuit depth $L\in\{8,12,16,20,24\}$ , for different spectral cutoffs $K\in\{0,2,4,6,8\}$ . Solid lines with markers show the RMSE computed over $2000$ i.i.d. samples $[U_{1}],\ldots,[U_{N}]\sim\mu_{\mathcal{PU}(2)}$ , while dashed lines show the fitted theoretical rate from Theorem 4.2 .

Figure 2. Empirical $\operatorname{RMSE}_{K,L}$ ( 5.10 ) between $f$ and its approximation $\mathcal{Q}_{K,L}f$ depending on the maximal circuit depth $L\in\{8,12,16,20,24,28\}$ , for different spectral cutoffs $K\in\{0,1,2,3,4\}$ . Solid lines with markers show the RMSE computed over $2000$ i.i.d. samples $[x_{1}],\ldots,[x_{N}]\sim\mu_{\mathbb{CP}^{1}}$ , while dashed lines show the fitted theoretical rate from Theorem 4.4 .

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Sep 5, 2026J/K move · Enter open · S save

Hierarchical Fourier Approximation for Variational Quantum Distribution Learning

Taha Hoseinpour Asli, Sajjad Hashemian, Ebrahim Ardeshir-Larijani

Sharif University of Technology · University of Tehran · Iran University of Science and Technology

Quantum Circuit Born Machines Quantum Machine Learning Quantum Generative Modeling Variational Quantum Circuits

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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.

Abstract

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