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Computational Mathematics and Mathematical Physics

On Comparison of Quantum Control Landscapes with Different Trapping Features

Alexander Nikolaevich Pechen

Published Aug 1, 2026• 0 citations
Publisher

Abstract

In this work, we consider a general method for numerically comparing the efficiency of optimization over control landscapes with different trapping features, both for unconstrained (using GRAPE) and constrained (using gradient projection method) optimization. Traps are controls which locally appear optimal while not being globally optimal. Such controls can significantly affect the efficiency of finding controls close to optimal with local search, e.g. gradient-based, algorithms. As an explicit example, we apply the general method to study unconstrained control landscapes for two three-level quantum $$\Xi $$ systems which are widely used in quantum optics and known to have different trapping features: a non-degenerate $$\Xi $$ -system for which the null control is a third-order trap, and a degenerate $$\Xi $$ -system for which the null control is a seventh-order trap. Using the general approach, we perform a comparative study of the impact of the null control on the efficiency of optimization and estimate the dependence of this impact on the maximal allowed number of iterations to check how the optimization is sensitive to this number. The results show that while quantitative behavior is different, the qualitative difference between the two control landscapes remains similar.

DOI
10.1134/s0965542526700570

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In this work, we consider a general method for numerically comparing the efficiency of optimization over control landscapes with different trapping features, both for unconstrained (using GRAPE) and constrained (using gradient projection method) optimization. Traps are controls which locally appear optimal while not being globally optimal. Such controls can significantly affect the efficiency of finding controls close to optimal with local search, e.g. gradient-based, algorithms. As an explicit example, we apply the general method to study unconstrained control landscapes for two three-level quantum Ξ\Xi Ξ systems which are widely used in quantum optics and known to have different trapping features: a non-degenerate Ξ\Xi Ξ -system for which the null control is a third-order trap, and a degenerate Ξ\Xi Ξ -system for which the null control is a seventh-order trap. Using the general approach, we perform a comparative study of the impact of the null control on the efficiency of optimization and estimate the dependence of this impact on the maximal allowed number of iterations to check how the optimization is sensitive to this number. The results show that while quantitative behavior is different, the qualitative difference between the two control landscapes remains similar.
In this work, we consider the problem of the analysis of quantum control landscapes in both situations when the controls during optimization are unconstrained and when they should satisfy a given constraint, for example their amplitude should be bounded by some constant value. For the analysis of unconstrained quantum control landscapes, a special approach was developed in [Pechen and Tannor, Israel J. of Chemistry 52 (5), 467–472 (2012)] based on a fixed step gradient ascent pulse engineering (GRAPE) algorithm. Here we also adopt a GRAPE approach based on the Pontryagin maximum principle (PMP). For the analysis of the constrained quantum control landscapes, we adopt the gradient projection method which was developed for quantum systems in [A. Oza et al., J. Phys. A: Math. Theor. 42, 205305 (2009)] (GPM; for our purpose the most suitable is a one-step GPM with a fixed step), both for closed and open quantum systems. As an explicit example, we apply these approaches to study quantum control landscape for optimization of an observable mean value for a three-level degenerate Ξ\XiΞ -system system having a seventh-order trap.
In this work, we perform an analytical and numerical analysis of quantum landscapes for controlling special four-level quantum systems for which we prove that the null control is a five-order trap: a V−VV-VV−V system and an anharmonic system. As a control goal, an observable optimization is considered. The rigorous theoretical analysis is followed by the numerical experiments based on the GRadient Ascent Pulse Engineering (GRAPE) algorithm and Gradient Projection Method (GPM), performed to investigate the behavior of the efficiency of optimization for unconstrained (using GRAPE) and constrained (using GPM) controls. As the main result, we observe an interesting phenomenon with a diverse behavior of the optimization efficiency depending on the system Hamiltonian -- sharp increase of the optimization efficiency up to 100% at certain distance from the null control for a V-V system, while much slower and less significant increase (and even small decrease) for a system with the chain interaction. This sharp difference might be related with the fine structure of the subspace of controls where second derivative of the objective functional is zero.
Methods of optimization and control of quantum systems are widely used for various applications in quantum technologies including laser control of chemical reactions, nuclear magnetic resonance, gate generation for quantum computing, describing quantum transport in photosynthesis, etc. Among many topics in quantum control, we briefly outline Riemannian optimization and gradient projection method (GPM) on complex Stiefel manifolds and their relation to quantum control, as well as various related optimization tools such as incoherent GRAPE (InGRAPE) modification of the gradient ascent pulse engineering (GRAPE) algorithm, GPM in the dynamic picture for constrained optimization, and demonstrate operation of GRAPE and GPM for an investigation of the optimization efficiency over unconstrained and constrained quantum control landscapes for an example of a three-level Ξ\Xi Ξ -system having a seventh-order trap.1
Quantum optimal control has enjoyed wide success for a variety of theoretical and experimental objectives. These favorable results have been attributed to advantageous properties of the corresponding control landscapes, which are free from local optima 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 full rank, and (3) the control field is not constrained. This paper explores how gradient searches for globally optimal control fields are affected by deviations from assumption (2). In some quantum control problems, so-called singular critical points, at which the Jacobian is rank deficient, may exist on the landscape. Using optimal control simulations, we show that search failure is only observed when a singular critical point is also a second-order trap, which occurs if the control problem meets additional conditions involving the system Hamiltonian and/or the control objective. All known second-order traps occur at constant control fields, and we also show that they only affect searches that originate very close to them. As a result, even when such traps exist on the control landscape, they are unlikely to affect well-designed gradient optimizations under realistic searching conditions.
The broad success of optimally controlling quantum systems with external fields has been attributed to the favorable topology of the underlying control landscape, where the landscape is the physical observable as a function of the controls. The control landscape can be shown to contain no suboptimal trapping extrema upon satisfaction of reasonable physical assumptions, but this topological analysis does not hold when significant constraints are placed on the control resources. This work employs simulations to explore the topology and features of the control landscape for pure-state population transfer with a constrained class of control fields. The fields are parameterized in terms of a set of uniformly spaced spectral frequencies, with the associated phases acting as the controls. This restricted family of fields provides a simple illustration for assessing the impact of constraints upon seeking optimal control. Optimization results reveal that the minimum number of phase controls necessary to assure a high yield in the target state has a special dependence on the number of accessible energy levels in the quantum system, revealed from an analysis of the first- and second-order variation of the yield with respect to the controls. When an insufficient number of controls and/or a weak control fluence are employed, trapping extrema and saddle points are observed on the landscape. When the control resources are sufficiently flexible, solutions producing the globally maximal yield are found to form connected "level sets" of continuously variable control fields that preserve the yield. These optimal yield level sets are found to shrink to isolated points on the top of the landscape as the control field fluence is decreased, and further reduction of the fluence turns these points into suboptimal trapping extrema on the landscape. Although constrained control fields can come in many forms beyond the cases explored here, the behavior found in this paper is illustrative of the impacts that constraints can introduce.
In this work, we consider a formulation of the optimal gate generation problem for a fluxonium model of superconducting systems and apply it to the numerical analysis of the corresponding quantum control landscape in a vicinity of the null control. The model includes Schrödinger equation with control and maximization of the objective functionals for gate generation and observable mean value. For a qutrit fluxonium model, the third level can be considered as one of the three computational states. We exploit gradient of the control objective and GRAPE algorithm to perform numerical investigation of the quantum control landscapes for generation of the Hadamard, T\mathrm{T}T , and S\mathrm{S}S gates and for maximizing mean value of a quantum observable in a vicinity of the null control. The obtained results show high efficiency of gradient-based optimization when it starts from the null control and decrease of the efficiency with increase of the distance of the initial control from the null control for the gate generation problems, and an opposite behaviour for the mean value maximization. For gate generation, the obtained behavior differs (is opposite) from the behavior found previously for maximizing mean value of an observable for Λ\LambdaΛ -type systems, that may indicate either the absence or low influence of trap at the null control for a fluxonium.