Control landscapes for two-level open quantum systems

TL;DR

This paper analyzes the control landscapes of two-level open quantum systems, revealing their topological properties and critical points, which explain the ease of achieving optimal control in practical quantum experiments.

Contribution

It provides a detailed characterization of the critical points and optimal solution manifolds for quantum control landscapes involving Kraus maps, including explicit structures and counts.

Findings

No local maxima or minima (false traps) in the landscape.

Existence of multi-dimensional sub-manifolds of global optima.

Critical points depend on initial system states, with specific saddle configurations.

Abstract

A quantum control landscape is defined as the physical objective as a function of the control variables. In this paper the control landscapes for two-level open quantum systems, whose evolution is described by general completely positive trace preserving maps (i.e., Kraus maps), are investigated in details. The objective function, which is the expectation value of a target system operator, is defined on the Stiefel manifold representing the space of Kraus maps. Three practically important properties of the objective function are found: (a) the absence of local maxima or minima (i.e., false traps); (b) the existence of multi-dimensional sub-manifolds of optimal solutions corresponding to the global maximum and minimum; and (c) the connectivity of each level set. All of the critical values and their associated critical sub-manifolds are explicitly found for any initial system state. Away from the absolute extrema there are no local maxima or minima, and only saddles may exist, whose number and the explicit structure of the corresponding critical sub-manifolds are determined by the initial system state. There are no saddles for pure initial states, one saddle for a completely mixed initial state, and two saddles for other initial states. In general, the landscape analysis of critical points and optimal manifolds is relevant to the problem of explaining the relative ease of obtaining good optimal control outcomes in the laboratory, even in the presence of the environment.