Optimal control for state preparation in two-qubit open quantum systems driven by coherent and incoherent controls via GRAPE approach

Abstract

In this work, we consider a model of two qubits driven by coherent and incoherent time-dependent controls. The dynamics of the system is governed by a Gorini–Kossakowski–Sudarshan–Lindblad master equation, where coherent control enters into the Hamiltonian and incoherent control enters into both the Hamiltonian (via Lamb shift) and the dissipative superoperator. We consider two physically different classes of interaction with coherent control and study the optimal control problem of state preparation formulated as minimization of the Hilbert–Schmidt distance’s square between the final density matrix and a given target density matrix at some fixed target time. Taking into account that incoherent control by its physical meaning is a non-negative function of time, we derive an analytical expression for the gradient of the objective and develop optimization approaches based on adaptation for this problem of GRadient Ascent Pulse Engineering (GRAPE). We study evolution of the von Neumann entropy, purity, and one-qubit reduced density matrices under optimized controls and observe a significantly different behavior of GRAPE optimization for the two classes of interaction with coherent control in the Hamiltonian.

On Reachable and Controllability Sets for Minimum-Time Control of an Open Two-Level Quantum System

“…The described below GPMs operate in theory with such controls, and the performed computer implementations of GPMs use piecewise linear interpolation for controls. For the non-differentiable objectives, we consider piecewise linear controls that, in contrast to piecewise constant controls used in the GRAPE-type method in [25], is another way of parameterization of controls.…”

“…However, in some cases, one can use the environment as a useful control resource, such as, for example, in the incoherent control approach [21,22], where the spectral, generally time-dependent and non-equilibrium density of incoherent photons is used as a control function jointly with the coherent control via lasers to manipulate such a quantum system dynamics. Following this approach, various types and aspects of optimal control problems for one-and two-qubit systems were analyzed [23][24][25][26][27][28].…”

“…• For finite-dimensional optimization under various classes of parameterized controls, e.g., gradient ascent pulse engineering GRAPE-type methods (e.g., [25,26,62], [27, Section 3]) (GRAPE-type methods operate with piecewise-constant controls, matrix exponentials, and gradients), CRAB ansatz [46,63] (coherent control is considered in terms of sine, cosine, etc. ), genetic algorithm (GA) [21,64,65], dual annealing [24], etc.…”

GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

“…Numerical simulations using expressions for the gradient and the evolution for one qubit were performed in [67] for the unitary gate generation problem, but without analytical analysis as it is done in this paper. Some problems for the two-qubit case were analyzed numerically in [68].…”

“…Numerical simulations using expressions for gradient and evolution for one qubit were performed in [85] for the unitary gate generation problem, but without analytical analysis as it is done in this paper. Some problems for the two-qubit case were analyzed numerically in [86]. An important problem is to analyze robustness of controls to various fluctuations and imperfections [87][88][89][90][91][92][93][94][95][96][97][98][99].…”