This paper addresses the nature and multiplicity of an optimally designed electric field for controlling quantum-dynamical processes. A rather general cost functional is considered, with only mild conditions called for amongst the various operators involved. An explicit upper bound on the magnitude of the controlling electric field is attained in terms of the norms of various operators entering into the control cost functional. An earlier work employing first-order perturbation theory arguments showed that, under rather mild assumptions, a denumerably infinite number of control-field solutions exists for the optimal control problem. In the present work, it is shown that through a bound on the remainder of the nonlinear terms in the expansion, this same conclusion concerning the control-field multiplicity continues to hold.
This work considers optimal control of quantum-mechanical systems within the framework of perturbation theory with respect to the controlling optical electric field. The control problem is expressed in terms of a cost functional including the physical objective, the penalties, and constraints. The resultant nonlinear variational equations are linearized by considering the lowest-order term in an expansion in powers of the optical-field strength. The optical field is found to satisfy a linear integral equation, and the solution may be expressed in terms of a generalized eigenvalue problem associated with the corresponding kernel. A full determination of the field is specified through the solution to the integral equation and the roots of an accompanying linearized spectral equation for a characteristic multiplier parameter. Each discrete value of the latter parameter corresponds to a particular solution to the variational equations. As a result, it is argued that under very general conditions there will be a denumerably infinite number of solutions to well-posed quantum-mechanical optimal-control problems.
This work presents a general framework for assessing the quality and robustness of control over quantum dynamics induced by an optical field
E(t). The control process is expressed in terms of a cost functional, including the physical objectives, penalties, and constraints. The first variations of such cost functionals have traditionally been utilized to create designs for the controlling electric fields. Here, the second variation of the cost functional is analyzed to explore (i) whether such solutions are locally optimal, and (ii) their degree of robustness. Both issues may be assessed from the eigenvalues of the stability operator S whose kernel $K(t,\ensuremath{\tau})$ is related to $\ensuremath{\delta}\mathcal{E}(t)/\ensuremath{\delta}\mathcal{E}(\ensuremath{\tau}){|}_{c}$ for
0<t, $\ensuremath{\tau}<~T,$ where
T is the target control time. Here
c denotes the constraint that the field satisfies the optimal control dynamical equations. The eigenvalues \ensuremath{\sigma} of S satisfying $\ensuremath{\sigma}<1$ assure local optimality of the control solution, with $\ensuremath{\sigma}=1$ being the critical value separating optimal solutions from false solutions (i.e., those with negative second variational curvature of the cost functional). In turn, the maximally robust control solutions with the least sensitivity to field errors also correspond to $\ensuremath{\sigma}=1.$ Thus, sufficiently high sensitivity of the field at one time
t to the field at another time \ensuremath{\tau} (i.e., $\ensuremath{\sigma}>1$) will lead to a loss of local optimality. An expression is obtained for a bound on the stability operator, and this result is employed to qualitatively analyze control behavior. From this bound, the inclusion of an auxiliary operator (i.e., other than the target operator) is shown to act as a stabilizer of the control process. It is also shown that robust solutions are expected to exist in both the strong- and weak-field regimes.
We investigate the problem of control of quantum systems. Various notions of controllability of quantum systems are discussed and Lie algebraic techniques are employed to derive necessary and sufficient conditions for the resulting degrees of controllability. The results are employed to study the degree of controllability for various model systems. The problem of optimal control is formulated as a problem of maximisation of the expectation value of an observable of the system subject to dynamical constraints and costs. A class of iterative algorithms is discussed, and shown to converge to a solution of the Euler-Lagrange equations, which is a necessary condition for optimality of a solution. A numerical realisation of the algorithm is presented and potential implementation of the resulting control fields in the laboratory is discussed. The algorithm is applied to find optimal control fields for a variety of applications including selective and simultaneous excitation of individual quantum dots in a globally addressed ensemble, creation of superposition states, and selective excitation of certain vibrational modes of a simple molecule such as Hydrogen Fluoride. In each case we systematically explore the parameter space of the algorithm to select solutions with desirable physical features from the multitude of possible choices. As there are many possible optimal control fields which give similar yields the question of whether some are more robust than others is investigated. A number of optimal fields are analysed quantitatively in terms of robustness with regard to temporal and spectral noise affecting the fields, uncertainty in the model parameters and the effect of experimental limitations such as restrictions on the bandwidth of the fields. It is shown that whilst the solutions tend to be sensitive to uncertainty in the energy levels chosen for the model, they appear to be surprisingly tolerant to even severe bandwidth limitations.
Optimally-shaped electromagnetic fields have the capacity to coherently control the dynamics of quantum systems and thus offer a promising means for controlling molecular transformations relevant to chemical, biological, and materials applications.Currently, advances in this area are hindered by the prohibitive cost of the quantum dynamics simulations needed to explore the principles and possibilities of molecular control.However, the emergence of nascent quantum-computing devices suggests that efficient simulations of quantum dynamics may be on the horizon.In this article, we study how quantum computers could be employed to design optimally-shaped fields to control molecular systems.We introduce a hybrid algorithm that utilizes a quantum computer for simulating the field-induced quantum dynamics of a molecular system in polynomial time, in combination with a classical optimization approach for updating the field.Qubit encoding methods relevant for molecular control problems are described, and procedures for simulating the quantum dynamics and obtaining the simulation results are discussed.Numerical illustrations are then presented that explicitly treat paradigmatic vibrational and rotational control problems, and also consider how optimally-shaped fields could be used to elucidate the mechanisms of energy transfer in light-harvesting complexes.Resource estimates, as well as a numerical assessment of the impact of hardware noise and the prospects of near-term hardware implementations, are provided for the latter task.
This paper presents a method for performing approximate optimal control simulations for quantum systems with multiple coupled degrees of freedom. The time evolution is simulated using the first-order Magnus expansion in the interaction picture, where the couplings between different degrees of freedom are treated as the perturbation. A numerical implementation procedure is presented that leverages upon pairwise couplings and the separability of the zeroth-order time evolution operator to achieve a reduced computational cost, which is analyzed with respect to the number of degrees of freedom. The formulation is compatible with gradient-free methods to optimize the control field, and a stochastic hill climbing algorithm is adopted for this purpose. As illustrations, optimal control simulations are performed for systems of two and three dipole-dipole-coupled molecular rotors under the influence of a control field. For the two-rotor system, the field is optimized to achieve either orientation or entanglement objectives. For the three-rotor system, the field is optimized either to orient all three rotors in the same direction or to orient one rotor in a particular direction while the other two rotors point in the opposite direction.
We consider the optimal control of quantum systems interacting nonlinearly with an electromagnetic field. We propose monotonically convergent algorithms to solve the optimal equations. The monotonic behavior of the algorithm is ensured by a nonstandard choice of the cost, which is not quadratic in the field. These algorithms can be constructed for pure- and mixed-state quantum systems. The efficiency of the method is shown numerically for molecular orientation with a nonlinearity of order 3 in the field. Discretizing the amplitude and the phase of the Fourier transform of the optimal field, we show that the optimal solution can be well approximated by pulses that could be implemented experimentally.