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arXiv:2211.02517 · International Journal of Modern Physics A

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

Vadim N. Petruhanov, Alexander Nikolaevich Pechen

Published Jul 9, 2022• 13 citations
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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.

DOI
10.1142/s0217751x22430175

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Известия Российской академии наук Серия математическая•Jan 1, 2023•0 citations
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.
Abstract In this work, we consider a pair of qubits controlled by coherent and incoherent controls. The dynamics of the two-qubit system is driven by a Gorini–Kossakowsky–Sudarchhan–Lindblad master equation where coherent control enters into the Hamiltonian and incoherent control inters into both the Hamiltonian (via Lamb shift) and the dissipative superoperator. Two classes of interaction between the system and the coherent field are considered. For this system, we analyze the control problem of generating a given target density matrix which is formulated as minimizing the Hilbert–Schmidt distance between the final density matrix and the target density matrix. Incoherent control is modeled as a sum of constant in time Gaussians with centers related with the transitions frequencies between the energy levels of the qubits. Coherent control in general formulation is considered as measurable function and in numerical experiments as piecewise constant function with constraints on magnitudes and variations. Finite-dimensional numerical optimization is performed using the dual annealing method; the corresponding results are described for some initial and target density matrices and for some set of the parameters of the control problem.
Abstract The GRadient Ascent Pulse Engineering (GRAPE) method is widely used for optimization in quantum control. GRAPE is gradient search method based on exact expressions for gradient of the control objective. It has been applied to various coherently controlled closed and open quantum systems. In this work, we adopt the GRAPE method for optimizing objective functionals in open quantum systems driven by both coherent and incoherent controls. In our case, a tailored or engineered environment acts on the controlled system as control via time-dependent decoherence rates γ i ( t ) or, equivalently, via spectral density n ω ( t ) of the environment . To develop the GRAPE approach for this problem, we compute gradient of various objectives for general N -level open quantum systems for the piecewise constant class of control. The case of a single qubit is considered in details and solved analytically. For this case, an explicit analytical expression for evolution and objective gradient is obtained via diagonalization of a 3 × 3 matrix determining the system’s dynamics in the Bloch ball. The diagonalization is obtained by solving a cubic equation via Cardano’s method. The efficiency of the algorithm is demonstrated through numerical simulations for the state-to-state transition problem and its complexity is estimated. Robustness of the optimal controls is also studied.
This work considers two-qubit open quantum systems driven by coherent and incoherent controls. Incoherent control induces time-dependent decoherence rates via time-dependent spectral density of the environment which is used as a resource for controlling the system. The system evolves according to the Gorini–Kossakowski– Sudarshan–Lindblad master equation with time-dependent coefficients. For two types of interaction with coherent control, three types of objectives are considered: 1) maximizing the Hilbert–Schmidt overlap between the final and target density matrices; 2) minimizing the Hilbert–Schmidt distance between these matrices; 3) steering the overlap to a given value. For the first problem, we develop the Krotov type methods directly in terms of density matrices with or without regularization for piecewise continuous controls with constaints and find the cases where the methods produce (either exactly or with some precision) zero controls which satisfy the Pontryagin maximum principle and produce the overlap’s values close to their upper bounds. For the problems 2) and 3), we find cases when the dual annealing method steers the objectives close to zero and produces a non-zero control.
We consider an open qutrit system in which the evolution of the density matrix ρ(t)\rho(t)ρ(t) is governed by the Gorini–Kossakowski–Sudarshan–Lindblad master equation with simultaneous coherent (in the Hamiltonian) and incoherent (in the dissipation superoperator) controls. To control the qutrit, we propose to use not only coherent control but also generally time-dependent decoherence rates which are adjusted by the so-called incoherent control. In our approach, the incoherent control makes the decoherence rates time-dependent in a specific controlled manner and within a clear physical mechanism. We consider the problem of maximizing the Hilbert–Schmidt overlap between the final state ρ(T)\rho(T)ρ(T) of the system and a given target state ρtarget\rho_{\text{target}}ρtarget​ , as well as the problem of minimizing the squared Hilbert–Schmidt distance between these states. For both problems, we perform their realifications, derive the corresponding Pontryagin functions, adjoint systems (with two variants of transversality conditions for the two terminal objectives), and gradients of the objectives, and adapt the one-, two-, and three-step gradient projection methods. For the problem of maximizing the overlap, we also adapt the regularized first-order Krotov method. In the numerical experiments, we analyze first the operation of the methods and second the obtained control processes, in respect of considering the environment as a resource via incoherent control.
This paper considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem of generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [48] for generic NNN-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage duration can be significantly less than the unmodified first stage duration, but at the cost of performing optimization in the class of piecewise constant controls.
This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem for generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [A. Pechen, Phys. Rev. A., 84, 042106 (2011)] for generic NNN-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system's state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage's duration can be significantly less than the unmodified first stage's duration, but at the cost of performing optimization in the class of piecewise constant controls. Bibliography: 72 titles.