Volume Fractions of the Kinematic "Near-Critical" Sets of the Quantum Ensemble Control Landscape

Abstract

An estimate is derived for the volume fraction of a subset $C_{\hat{I}}{\mu }^P=\{U:||gradJ(U)|\le \hat{I}\mu \}\subset \mathrm{U}(N)$ in the neighborhood of the critical set $C^P\simeq \mathrm{U}(\mathbf{n})P\mathrm{U}(\mathbf{m})$ of the kinematic quantum ensemble control landscape J(U) = Tr(UρU' O), where $U$ represents the unitary time evolution operator, ρ is the initial density matrix of the ensemble, and O is an observable operator. This estimate is based on the Hilbert-Schmidt geometry for the unitary group and a first-order approximation of $||gradJ(U)|{|}^2$. An upper bound on these near-critical volumes is conjectured and supported by numerical simulation, leading to an asymptotic analysis as the dimension $N$ of the quantum system rises in which the volume fractions of these "near-critical" sets decrease to zero as $N$ increases. This result helps explain the apparent lack of influence exerted by the many saddles of $J$ over the gradient flow.

27 pages, 1 figure