Characterization of the Critical Submanifolds in Quantum Ensemble Control Landscapes

TL;DR

This paper introduces a method to analyze the topology of quantum control landscapes by linking critical submanifolds to contingency tables based on eigenvalue degeneracies, aiding understanding of quantum control optimization.

Contribution

It presents a novel scheme to compute the critical topology of quantum ensemble control landscapes using contingency tables related to eigenvalue degeneracies.

Findings

Critical submanifolds correspond to contingency tables.

Landscape characteristics depend on eigenvalue degeneracies.

Method effectively illustrates landscape topology with examples.

Abstract

A quantum control landscape is defined as the physical objective as a function of the control variables to be optimized. Analyzing the topology of these landscapes is important for understanding the origins of the increasing number of laboratory successes in the optimal control of quantum processes. This paper proposes a simple scheme to compute the characteristics of the critical topology of the quantum ensemble control landscapes for observables, showing that the set of disjoint critical submanifolds one-to-one corresponds to a finite number of contingency tables that solely depend on the degeneracy structure of the eigenvalues of the initial system density matrix and the observable to be controlled. The landscape characteristics can be calculated as functions of the table entries, including the dimensions and the numbers of positive and negative eigenvalues of the Hessian quadratic form of each of the connected components of the critical submanifolds. Typical examples are given to illustrate the effectiveness of this method.