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

TL;DR

This paper estimates the volume of near-critical sets in quantum control landscapes, showing they diminish as system size grows, which explains the limited impact of saddle points on gradient-based optimization.

Contribution

It provides a novel geometric estimate for near-critical set volumes in quantum control landscapes and analyzes their asymptotic behavior as system dimension increases.

Findings

Near-critical volumes decrease to zero with increasing system size

Supports the idea that saddle points have limited influence on gradient flow

Uses Hilbert-Schmidt geometry and numerical simulations for analysis

Abstract

An estimate is derived for the volume fraction of a subset $C_{\epsilon}^{P} = \{U : ||grad J(U)|\leq {\epsilon}\}\subset\mathrm{U}(N)$ 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(U\rho U' O), where $U$ represents the unitary time evolution operator, {\rho} 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 $||grad J(U)||^2$. 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.