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This paper establishes a rigorous mathematical link between the Krylov hierarchy of quantum Fisher information and the Lanczos algorithm, providing new insights into convergence regimes for spectral estimation in noisy intermediate-scale quantum (NISQ) devices. By distinguishing between computational Hilbert-Schmidt geometry and state-dependent geometries, it clarifies when exponential versus algebraic decay occurs during iterative optimization protocols. These findings offer a theoretical foundation for improving parameter estimation efficiency without requiring full tomography of complex many-body states.
Krylov Distribution and Spectral Convergence of Quantum Fisher Information
arXiv:2602.19750v3 Announce Type: replace Abstract: We establish a direct correspondence between the Krylov hierarchy of quantum Fisher information (QFI) and a Krylov-resolved decomposition of the symmetric logarithmic derivative (SLD). Building on the existing Krylov formulation of QFI, we show that the metrological information resolved at each Krylov level is exactly encoded in the incremental gain of the Lanczos hierarchy, giving the Krylov distribution a direct information-theoretic interpretation. A dual spectral formulation distinguishes the Hilbert--Schmidt geometry used to construct the computational Lanczos hierarchy from the state-dependent geometry in which the SLD and QFI are naturally defined, and relates their associated Liouville measures. This distinction clarifies the origin of different convergence regimes: we recover exponential convergence for gapped spectra and derive an exact algebraic hard-edge convergence law for the Jacobi class, including the exact finite-order error and the marginal case separating finite from divergent QFI.
Abstract:We establish a direct correspondence between the Krylov hierarchy of quantum Fisher information (QFI) and a Krylov-resolved decomposition of the symmetric logarithmic derivative (SLD). Building on the existing Krylov formulation of QFI, we show that the metrological information resolved at each Krylov level is exactly encoded in the incremental gain of the Lanczos hierarchy, giving the Krylov distribution a direct information-theoretic interpretation. A dual spectral formulation distinguishes the Hilbert--Schmidt geometry used to construct the computational Lanczos hierarchy from the state-dependent geometry in which the SLD and QFI are naturally defined, and relates their associated Liouville measures. This distinction clarifies the origin of different convergence regimes: we recover exponential convergence for gapped spectra and derive an exact algebraic hard-edge convergence law for the Jacobi class, including the exact finite-order error and the marginal case separating finite from divergent QFI.
12 pages, 2 figures. V2: Minor corrections, conceptual framework clarified, presentation and numerical results improved, refs added, 17 pages, 3 figures
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th)
arXiv:2602.19750 [quant-ph]
(or arXiv:2602.19750v3 [quant-ph] for this version)
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