Search papers, labs, and topics across Lattice.
This paper leverages harmonic analysis on compact groups and homogeneous spaces to derive approximation rates for Sobolev functions on quantum state and operation spaces. By characterizing Sobolev spaces via irreducible representations and the Laplace-Beltrami operator, it enables a principled decomposition of approximation error into spectral truncation and finite-depth quantum circuit components. The key result establishes quantitative rates that disentangle these two error sources, providing a theoretical foundation for approximating complex quantum functions with circuits.
Quantum circuits can approximate complex Sobolev functions on quantum state and operation spaces with provable, disentangled spectral and circuit approximation errors.
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.
