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arXiv:1807.07450 · Physical Review A

Variational approach to the optimal control of coherently driven, open quantum system dynamics

Vasco Cavina, Andrea Mari, A. Carlini, Vittorio Giovannetti

Published Nov 20, 2018• 19 citations
arXivHTMLPDF

Abstract

Quantum coherence inherently affects the dynamics and the performances of a quantum machine. Coherent control can, at least in principle, enhance the work extraction and boost the velocity of evolution in an open quantum system. Using advanced tools from the calculus of variations and reformulating the control problem in the instantaneous Hamiltonian eigenframe, we develop a general technique for minimizing a wide class of cost functionals when the external control has access to full rotations of the system Hamiltonian. The method is then applied both to time and heat loss minimization problems and explicitly solved in the case of a two-level system in contact with either bosonic or fermionic thermal environments.

DOI
10.1103/physreva.98.052125

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Quantum coherence inherently affects the dynamics and the performances of a quantum machine. Coherent control can, at least in principle, enhance the work extraction and boost the velocity of evolution in an open quantum system. Using advanced tools from the calculus of variations and reformulating the control problem in the instantaneous Hamiltonian eigenframe, we develop a general technique for minimizing a wide class of cost functionals when the external control has access to full rotations of the system Hamiltonian. The method is then applied both to time and heat loss minimization problems and explicitly solved in the case of a two-level system in contact with either bosonic or fermionic thermal environments.
This paper is concerned with variational methods for nonlinear open quantum systems with Markovian dynamics governed by Hudson-Parthasarathy quantum stochastic differential equations. The latter are driven by quantum Wiener processes of the external boson fields and are specified by the system Hamiltonian and system-field coupling operators. We consider the system response to perturbations of these energy operators and introduce a transverse Hamiltonian which encodes the propagation of the perturbations through the unitary system-field evolution. This provides a tool for the infinitesimal perturbation analysis and development of optimality conditions for coherent quantum control problems. We apply the transverse Hamiltonian variational technique to a mean square optimal coherent quantum filtering problem for a measurement-free cascade connection of quantum systems.
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.
We apply advanced methods of control theory to open quantum systems and we determine finite-time processes which are optimal with respect to thermodynamic performances. General properties and necessary conditions characterizing optimal drivings are derived, obtaining bang-bang-type solutions corresponding to control strategies switching between adiabatic and isothermal transformations. A direct application of these results is the maximization of the work produced by a generic quantum heat engine, where we show that the maximum power is directly linked to a particular conserved quantity naturally emerging from the control problem. Finally we apply our general approach to the specific case of a two-level system, which can be put in contact with two different baths at fixed temperatures, identifying the processes that minimize heat dissipation. Moreover, we explicitly solve the optimization problem for a cyclic two-level heat engine driven beyond the linear-response regime, determining the corresponding optimal cycle, the maximum power, and the efficiency at maximum power.
The article considers a two-level open quantum system evolving under the action of coherent and incoherent controls using the general control method proposed in Phys. Rev. A. 73, 062102 (2006). Coherent control determines the Hamiltonian aspects of the dynamics whereas incoherent control determines the dissipative aspects. The goal is to find controls which steer the initial density matrix into a state which maximizes overlap with a predefined target density matrix. The controlled dynamics is represented as evolution in the Bloch ball and is analyzed analytically using Pontryagin maximum principle and numerically using optimization either in the functional space of controls (with the conditional and projected gradient methods) or through reduction to a finite-dimensional control space.
Optimal control in open quantum systems is considered, with a focus on a two-level logical qubit system whose time evolution is governed by Markovian dynamics described within the Lindblad (GKSL) formalism. The objective is to minimize the net heat transferred to the environment during the transition while explicitly accounting for the work performed by the external control field. In this context, we formulate a variational optimal control framework for minimizing dissipated heat under fidelity constraints, which establishes existence under standard assumptions of bounded controls and lower semi-continuity of at least one bounded control that minimizes the environmental heat dissipation functional while maintaining the final-state fidelity above a designated threshold. The analysis combines the direct method of the calculus of variations with the Pontryagin Maximum Principle. The direct method in the calculus of variations (in L ∞ spaces) guarantees existence by demonstrating the compactness of admissible controls and the lower semi-continuity of the cost functional. Second, the Pontryagin Maximum Principle for open quantum systems establishes essential optimality conditions, resulting in coupled state–costate equations and switching functions that define optimal controls. We establish a physical foundation for the optimization problem by defining internal energy, heat flow, and work within the Lindblad framework, resulting in a thermodynamically coherent expression for dissipated energy. This formulation allows for both analytical understanding together with a practical numerical approximation based on the MDR-GRAPE algorithm. To demonstrate numerical consistency, we present MDR-GRAPE, a modified gradient-based algorithm that extends GRAPE to dissipative environments, integrating positivity-preserving techniques for density matrices. Numerical experiments demonstrate reproducible convergence across the five independent initializations considered in this study, providing empirical evidence of consistent convergence toward feasible low-dissipation solutions.