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arXiv:2607.01217 · Physica Scripta

Diverse efficiency of observable optimization for four-level quantum systems with higher-order traps

Alexander Pechen, Boris Volkov

quant-phSubmitted Jul 1, 2026• Announced Jul 2, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $V-V$ 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.

Journal reference
Physica Scripta Vol. 101, N. 2, Art. No. 025101 (2026)
DOI
10.1088/1402-4896/ae2f3d
Comments
This work is original preprint. Published version was significantly extended

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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.
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 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 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.
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
Abstract In this work, we adopt the gradient projection method (GPM) to problems of quantum control. For general N-level closed and open quantum systems, we derive the corresponding adjoint systems and gradients of the objective functionals and provide the projection versions of the Pontryagin maximum principle and the GPM, all directly in terms of quantum objects such as evolution operator, Hamiltonians, density matrices, etc. Various forms of the GPM, including one- and two-step, are provided and compared. We formulate the GPM both for closed and open quantum systems, latter for the general case with simultaneous coherent and incoherent controls. The GPM is designed to perform local gradient based optimization in the case when bounds are imposed on the controls. The main advantage of the method is that it allows to exactly satisfy the bounds, in difference to other approaches such as adding constraints as weight to an objective. We apply the GPM to several examples including generation of one- and two-qubit gates and two-qubit Bell and Werner states for models of superconducting qubits under the constraint when controls are zero at the initial and final times, steering an open quantum system to a target density matrix for simulating action of the Werner–Holevo channel, etc.
Abstract Mathematical analysis of quantum control landscapes, which aims to prove either absence or existence of traps for quantum control objective functionals, is an important topic in quantum control. In this work, we provide a rigorous analysis of quantum control landscapes for ultrafast generation of single-qubit quantum gates and show, combining analytical methods based on a sophisticated analysis of spectrum of the Hessian, and numerical optimization methods such as gradient ascent pulse engineering, differential evolution, and dual annealing, that control landscape for ultrafast generation of phase shift gates is free of traps.