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Journal of Mathematical Physics

The energy minimization problem for two-level dissipative quantum systems

Bernard Bonnard A, Olivier Cots, Nataliya Shcherbakova, Dominique Sugny

Published Sep 1, 2010• 20 citations
Publisher

Abstract

In this article, we study the energy minimization problem of dissipative two-level quantum systems whose dynamics is governed by the Kossakowski–Lindblad equations. In the first part, we classify the extremal curve solutions of the Pontryagin maximum principle. The optimality properties are analyzed using the concept of conjugate points and the Hamilton–Jacobi–Bellman equation. This analysis completed by numerical simulations based on adapted algorithms allows a computation of the optimal control law whose robustness with respect to the initial conditions and dissipative parameters is also detailed. In the final section, an application in nuclear magnetic resonance is presented.

DOI
10.1063/1.3479390

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In this article, we study the energy minimization problem of dissipative two-level quantum systems whose dynamics is governed by the Kossakowski–Lindblad equations. In the first part, we classify the extremal curve solutions of the Pontryagin maximum principle. The optimality properties are analyzed using the concept of conjugate points and the Hamilton–Jacobi–Bellman equation. This analysis completed by numerical simulations based on adapted algorithms allows a computation of the optimal control law whose robustness with respect to the initial conditions and dissipative parameters is also detailed. In the final section, an application in nuclear magnetic resonance is presented.
We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.
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.
The aim of this contribution is to re ne some of the computationsof [6]. The Lindblad equation modelling a two-level dissipative quantum system is investigated. The control can be interpretated as the action of a laser to rotate a molecule in gas phase, or as the e ect of a magnetic eld on a spin 1=2 particle. For the energy cost, normal extremals of the maximum principle are solution to a three-dimensional Hamiltonian with parameters. The analysis is focussed on an integrable submodel which de nes outside singularities a pseudo-Riemannian metric in dimension ve. Complete quadratures are given for this subcase by means of Weierstra elliptic functions. Preliminary computations of cut and conjugate loci are also provided for a two-dimensional restriction using [9].