Quantum Bernstein--Kolmogorov--Arnold approximation

@misc{dzhenzher2026quantum,
  title = {{Quantum Bernstein--Kolmogorov--Arnold approximation}},
  author = {Sviatoslav V. Dzhenzher},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41608},
  eprint = {2610.41608},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41608},
}

TY  - UNPB
TI  - Quantum Bernstein--Kolmogorov--Arnold approximation
AU  - Sviatoslav V. Dzhenzher
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41608
UR  - https://arxiv.org/abs/2610.41608
AB  - This paper addresses the expressivity and representation gap in Quantum Neural Networks by establishing a quantum analogue of the classical Kolmogorov--Arnold representation theorem. We propose a constructive approximation framework over fixed input coherent states, where the dependence of the function argument is embedded entirely in coordinate qubits, while the target function dictates a single outer unitary evolution. By applying a Sprecher-type representation to the rotation angle of the state, we demonstrate that the multidimensional operator factorises into 2d+1 commuting universal inner observables. We prove that the resulting measurement probability precisely reproduces the multidimensional Bernstein polynomial of the target function. For any continuous multivariate function, the joint quantum state converges uniformly to the target output register, linking classical superposition theory with probabilistic and coherent quantum circuits.
ER  -