The Optimal Control Landscape for the Generation of Unitary Transformations with Constrained Dynamics

TL;DR

This paper analyzes the control landscape for generating quantum unitary transformations, showing that certain symmetries do not introduce traps, thus broadening the scope of systems suitable for optimal control methods.

Contribution

It extends previous control landscape theory by demonstrating that specific Hamiltonian symmetries do not compromise the trap-free topology, enabling more systems to be controlled optimally.

Findings

Symmetries in Hamiltonians do not create traps in the control landscape.

The trap-free topology persists under certain dynamical constraints.

Control problems remain solvable by optimization heuristics despite symmetries.

Abstract

The reliable and precise generation of quantum unitary transformations is essential to the realization of a number of fundamental objectives, such as quantum control and quantum information processing. Prior work has explored the optimal control problem of generating such unitary transformations as a surface optimization problem over the quantum control landscape, defined as a metric for realizing a desired unitary transformation as a function of the control variables. It was found that under the assumption of non-dissipative and controllable dynamics, the landscape topology is trap-free, implying that any reasonable optimization heuristic should be able to identify globally optimal solutions. The present work is a control landscape analysis incorporating specific constraints in the Hamiltonian corresponding to certain dynamical symmetries in the underlying physical system. It is found that the presence of such symmetries does not destroy the trap-free topology. These findings expand the class of quantum dynamical systems on which control problems are intrinsically amenable to solution by optimal control.