Generalized monotonically convergent algorithms for solving quantum optimal control problems

Abstract

A wide range of cost functionals that describe the criteria for designing optimal pulses can be reduced to two basic functionals by the introduction of product spaces. We extend previous monotonically convergent algorithms to solve the generalized pulse design equations derived from those basic functionals. The new algorithms are proved to exhibit monotonic convergence. Numerical tests are implemented in four-level model systems employing stationary and/or nonstationary targets in the absence and/or presence of relaxation. Trajectory plots that conveniently present the global nature of the convergence behavior show that slow convergence may often be attributed to "trapping" and that relaxation processes may remove such unfavorable behavior.

Optimal control of time-dependent targets

“…For η, γ ∈ [0, 2] (similar to [15] and [19]), the iteration converges monotonically, i.e., δJ (k+1,k) ≥ 0. This iteration converges monotonically and quadratically in terms of the field deviations between two iterations.…”

“…While Ref. [19] presents the monotonically convergent algorithm, the full power of the method has not been exploited as yet. The challenge is the control of a truly time-dependent target represented by a positive-semidefinite, explicitly timedependent operator.…”

“…The third method is an optimal control scheme for time-dependent targets [19]. Basically, it combines optimal control schemes for time-independent targets [14,20] with an extension to Liouville-space [21].…”

Discrete-valued-pulse optimal control algorithms: Application to spin systems

“…In this paper, we consider specifically the GRAPE algorithm [25], but the same construction of the discrete version can be used for other algorithms such as the monotonic or Krotov ones [32,[51][52][53][54] (see also the general analysis of such methods [55,56]). …”

Optimal control of a charge qubit in a double quantum dot with a Coulomb impurity

“…The optimal field is the field employed in order to steer a dynamical system from a initial state to a desired target state minimizing a cost functional which generally penalizes the energy (fluence) of the pulse. A great effort has been invested in recent years in the development of different methods in order to solve the optimal equations [26,27,28,29,30]. Monotonically convergent iterative schemes proposed by Tannor et al [31] and Rabitz et al [32] have been successfully applied to the control of different quantum phenomena, mainly related to chemical process [33,34].…”