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

arXiv: https://arxiv.org/abs/2610.13477 · HTML: https://mlsanity.com/paper/2610.13477/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 36 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 5 of 5)

- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs](https://mlsanity.com/paper/2307.12904/) (2023)
- [Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits](https://mlsanity.com/paper/2310.07528/) (2023)
- [Breaking the cubic barrier in the Solovay-Kitaev algorithm](https://mlsanity.com/paper/2306.13158/) (2023)
- [Approximation rates of quantum neural networks for periodic functions via Jackson's inequality](https://mlsanity.com/paper/2511.16149/) (2025)

## Abstract (arXiv)

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.