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Harmonic analysis and circuit approximation of Sobolev functions on quantum input spaces
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.
Related
- Monotone Sobolev functions: approximation, critical points, and level sets
- Local Approximation of A-Harmonic Functions and the Super-Localization Conjecture
- Beyond Constant Truncations: The Bounded Sobolev Approximation Gap
- Galerkin Approximation of the Fractional Sobolev Constant
- Finite-Support Approximation in Variable-Coefficient Sobolev Spaces on Nonhomogeneous Flow-Measure Trees
BibTeX
@misc{schmocker2026harmonic,
title = {{Harmonic analysis and circuit approximation of Sobolev functions on quantum input spaces}},
author = {Philipp Schmocker},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13477},
eprint = {2610.13477},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13477},
}
RIS
TY - UNPB
TI - Harmonic analysis and circuit approximation of Sobolev functions on quantum input spaces
AU - Philipp Schmocker
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13477
UR - https://arxiv.org/abs/2610.13477
AB - 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.
ER -