Quantum Natural Gradient on Quotient Spaces

Abstract

A parametrized quantum circuit reports its state geometry through a quantum Fisher information matrix (QFIM), often singular. A small Fisher value can reflect exact state-preserving redundancy, compression by the circuit chart, or weak intrinsic distinguishability, and these mechanisms call for different numerical treatments. We show that the circuit metric factors as F=B^*MB, where B is the state-level circuit-to-orbit differential and M is the intrinsic Fisher operator on the reachable orbit. The factorization identifies the exact kernel as ker B, separates coordinate transfer from intrinsic geometry, and yields the condition for a circuit to realize an orbit-level quantum natural-gradient (QNG) direction. When the prescribed redundancy exhausts the Fisher kernel, the Moore--Penrose update is the minimum-norm horizontal lift of the quotient Riemannian gradient. At critical points with a locally diffeomorphic quotient-to-orbit map, chart singular values cancel from the linearized QNG operator while intrinsic anisotropy remains; in the trace-orthonormal full-control generator frame, excitation-gap anisotropy gives κ_QNG=κ_Eucl. Representation theory makes M explicit on highest-weight, Slater, and fermionic-Gaussian orbits, and cominuscule fidelity flow becomes integrable, with conserved principal-defect ratios, cubic Lie-retracted convergence at η=2, and stability boundary η=4. Finite data impose a second boundary: an estimated QFIM and its confidence radius alone cannot distinguish an exact zero from a small physical mode, so the estimated spectrum alone cannot license hard projection. Under depolarization, inverse-Fisher scaling amplifies mean updates and fluctuations together and cannot restore update signal-to-noise. A redundant Slater/Givens circuit confirms exact transfer identities and illustrates finite-shot tradeoffs.

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BibTeX RIS

BibTeX

@misc{chen2026quantum,
  title = {{Quantum Natural Gradient on Quotient Spaces}},
  author = {Zeyu Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.29350},
  eprint = {2608.29350},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29350},
}

RIS

TY  - UNPB
TI  - Quantum Natural Gradient on Quotient Spaces
AU  - Zeyu Chen
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2608.29350
UR  - https://arxiv.org/abs/2608.29350
AB  - A parametrized quantum circuit reports its state geometry through a quantum Fisher information matrix (QFIM), often singular. A small Fisher value can reflect exact state-preserving redundancy, compression by the circuit chart, or weak intrinsic distinguishability, and these mechanisms call for different numerical treatments. We show that the circuit metric factors as F=B^*MB, where B is the state-level circuit-to-orbit differential and M is the intrinsic Fisher operator on the reachable orbit. The factorization identifies the exact kernel as ker B, separates coordinate transfer from intrinsic geometry, and yields the condition for a circuit to realize an orbit-level quantum natural-gradient (QNG) direction. When the prescribed redundancy exhausts the Fisher kernel, the Moore--Penrose update is the minimum-norm horizontal lift of the quotient Riemannian gradient. At critical points with a locally diffeomorphic quotient-to-orbit map, chart singular values cancel from the linearized QNG operator while intrinsic anisotropy remains; in the trace-orthonormal full-control generator frame, excitation-gap anisotropy gives κ_QNG=κ_Eucl. Representation theory makes M explicit on highest-weight, Slater, and fermionic-Gaussian orbits, and cominuscule fidelity flow becomes integrable, with conserved principal-defect ratios, cubic Lie-retracted convergence at η=2, and stability boundary η=4. Finite data impose a second boundary: an estimated QFIM and its confidence radius alone cannot distinguish an exact zero from a small physical mode, so the estimated spectrum alone cannot license hard projection. Under depolarization, inverse-Fisher scaling amplifies mean updates and fluctuations together and cannot restore update signal-to-noise. A redundant Slater/Givens circuit confirms exact transfer identities and illustrates finite-shot tradeoffs.
ER  -

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