Characterization of the Critical Sets of Quantum Unitary Control Landscapes

TL;DR

This paper analyzes the critical points of quantum control landscapes related to unitary transformations, characterizing their structure and stability to inform quantum control optimization.

Contribution

It provides a detailed characterization of the critical sets and points of various quantum control landscapes, including their Morse-Bott properties and implications for dynamical landscapes.

Findings

Critical points include maxima, minima, and saddles.

Landscapes can be Morse-Bott functions on unitary groups.

Results inform optimization strategies in quantum control.

Abstract

This work considers various families of quantum control landscapes (i.e. objective functions for optimal control) for obtaining target unitary transformations as the general solution of the controlled Schr\"odinger equation. We examine the critical point structure of the kinematic landscapes J_F (U) = ||(U-W)A||^2 and J_P (U) = ||A||^4 - |Tr(AA'W'U)|^2 defined on the unitary group U(H) of a finite-dimensional Hilbert space H. The parameter operator A in B(H) is allowed to be completely arbitrary, yielding an objective function that measures the difference in the actions of U and the target W on a subspace of state space, namely the column space of A. The analysis of this function includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima, minima, and saddles. In addition, we consider the question of whether these landscapes are Morse-Bott functions on U(H). Landscapes based on the intrinsic (geodesic) distance on U(H) and the projective unitary group PU(H) are also considered. These results are then used to deduce properties of the critical set of the corresponding dynamical landscapes.