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.

Quantum Physics (quant-ph); Machine Learning (cs.LG); Representation Theory (math.RT); Spectral Theory (math.SP)

arXiv:2610.13477 [quant-ph]

(or arXiv:2610.13477v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.13477 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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