Gradient projection method for constrained quantum control

TL;DR

This paper introduces a Gradient Projection Method (GPM) tailored for quantum control problems, enabling efficient local optimization while exactly satisfying control bounds in both closed and open quantum systems.

Contribution

The work develops a GPM framework directly in quantum terms, including adjoint systems, gradients, and projection versions of optimality conditions, applicable to various quantum control scenarios.

Findings

GPM effectively handles control bounds in quantum optimization.

The method successfully generates quantum gates and states under constraints.

Applications include superconducting qubits and quantum channel simulations.

Abstract

In this work, we adopt the Gradient Projection Method (GPM) to problems of quantum control. For general $N$-level closed and open quantum systems, we derive the corresponding adjoint systems and gradients of the objective functionals, and provide the projection versions of the Pontryagin maximum principle and the GPM, all directly in terms of quantum objects such as evolution operator, Hamiltonians, density matrices, etc. Various forms of the GPM, including one- and two-step, are provided and compared. We formulate the GPM both for closed and open quantum systems, latter for the general case with simultaneous coherent and incoherent controls. The GPM is designed to perform local gradient based optimization in the case when bounds are imposed on the controls. The main advantage of the method is that it allows to exactly satisfy the bounds, in difference to other approaches such as adding constraints as weight to objective. We apply the GPM to several examples including generation of one- and two-qubit gates and two-qubit Bell and Werner states for models of superconducting qubits under the constraint when controls are zero at the initial and final times, and steering an open quantum system state to a target density matrix for simulating action of the Werner-Holevo channel, etc.