Searching for quantum optimal controls under severe constraints

Abstract

The success of quantum optimal control for both experimental and theoretical objectives is connected to the topology of the corresponding control landscapes, which are free from local traps if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is of full rank, and (3) there are no constraints on the control field. This paper investigates how the violation of assumption (3) affects gradient searches for globally optimal control fields. The satisfaction of assumptions (1) and (2) ensures that the control landscape lacks fundamental traps, but certain control constraints can still introduce artificial traps. Proper management of these constraints is an issue of great practical importance for numerical simulations as well as optimization in the laboratory. Using optimal control simulations, we show that constraints on quantities such as the number of control variables, the control duration, and the field strength are potentially severe enough to prevent successful optimization of the objective. For each such constraint, we show that exceeding quantifiable limits can prevent gradient searches from reaching a globally optimal solution. These results demonstrate that careful choice of relevant control parameters helps to eliminate artificial traps and facilitate successful optimization.

Searching for an optimal control in the presence of saddles on the quantum-mechanical observable landscape

“…The results of the simulations in Ref. [119] support this conclusion; for a control problem that lacks saddles, all optimizations were successful for τ ≤ 2 × 10 −3 . The searches on a landscape that has an attractive saddle require significantly more accurate solutions to Eq.…”

“…Previous simulations of observable control problems with gradient-based algorithms have regularly reached the landscape maximum value [110,119]. However, searches may converge more slowly while coming close to saddles, increasing the search effort.…”

“…(12), are a severe constraint that may result in search failure (i.e., a decrease in the value of the objective functional after an iteration). However, the simulations in [119] involved the state-transition landscape, which lacks saddles. When a gradient search is close to a critical point of the landscape, the norm of the gradient is small, and more accuracy may be required of the solutions to Eq.…”

“…(12) is determined by the error tolerance τ . In a previous numerical study that used a gradient algorithm to investigate the role of control constraints in OCT optimization, the effect of changing this tolerance was studied [119] and it was determined that a choice of τ = 10 −8 yields sufficiently accurate solutions to find optimal controls for the state-transition objective. It was also shown that large values of τ , and the resulting inaccurate solutions to Eq.…”

Controlling open quantum systems: tools, achievements, and limitations

“…More than one constraint may be employed at a time, with different weights allowing to emphasize importance of one compared to another [162,164]. While additional constraints provide information that guides the optimization algorithm, they also restrict the space of admissible solutions [165,166]. Therefore care needs to be taken to balance their benefits and their disadvantages.…”