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arXiv:2304.03355 · Russian Mathematical Surveys

Higher-order traps for some strongly degenerate quantum control systems

Boris Olegovich Volkov, Alexander Nikolaevich Pechen

Published Jan 1, 2023• 10 citations
arXivHTMLPDF

Abstract

Quantum control is necessary for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems. An important problem in quantum control is to establish whether the control objective functional has trapping behaviour or no, namely if it has or no traps -- controls from which it is difficult to escape by local search optimization methods. Higher order traps were previously introduced in [A. N. Pechen, D. J. Tannor, "Are there traps in quantum control landscapes?", Phys. Rev. Lett., 106 (2011), 120402], where 3-rd order traps were found. In this note we show that traps of arbitrarily high order exist for controllable quantum systems with special symmetry in the Hamiltonian.

DOI
10.4213/rm10069e

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Quantum control is necessary for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems. An important problem in quantum control is to establish whether the control objective functional has trapping behaviour or no, namely if it has or no traps -- controls from which it is difficult to escape by local search optimization methods. Higher order traps were previously introduced in [A. N. Pechen, D. J. Tannor, "Are there traps in quantum control landscapes?", Phys. Rev. Lett., 106 (2011), 120402], where 3-rd order traps were found. In this note we show that traps of arbitrarily high order exist for controllable quantum systems with special symmetry in the Hamiltonian.
Abstract An important problem in quantum control is the analysis of quantum control landscapes and its trapping features. The notion of a trap in quantum control landscape was introduced in [H. Rabitz, M. Hsieh, C. Rosenthal, Quantum optimally controlled transition landscapes, Science 303, 1998 (2004)]. Later, the notion of higher-order traps was introduced [A. Pechen and D.J. Tannor, Are there traps in quantum control landscapes?, Phys. Rev. Lett. 106, 120402 (2011)] and such higher-order traps were shown to exist for a variety of important applications for physical systems such as for example Λ-atoms widely used in quantum optics and superconducting transmon qubit systems used as one of the potential platform for quantum computation. In this note, we provide several results on the existence of higher-order traps in quantum control landscapes.
Abstract Quantum control is a necessary tool for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems for various tasks. Traps are points of local but not global optimum of the objective functional for a given quantum control problem. In a more general sense, traps are critical points of the objective functional which are hard to escape by local search algorithms. Here a review of some results of the analysis of possibility of having traps in landscapes of coherently controlled closed quantum systems is given. In one-qubit case, there are no traps. For special multilevel quantum systems, higher-order traps may appear.
Many quantum control problems are formulated as a search for an optimal field that maximizes a physical objective. This search is performed over a landscape defined as the objective as a function of the control field. A recent Letter [A. N. Pechen and D. J. Tannor, Phys. Rev. Lett. 106, 120402 (2011)] asserts that the existence of special landscape critical points (CPs) with trapping characteristics is "contrary to recent claims in the literature" and "can have profound implications for both theoretical and experimental quantum control studies." We show here that these assertions are inaccurate and misleading.
$\mathrm{\ensuremath{\Lambda}}$, VVV, $\mathrm{\ensuremath{\Xi}}$ (ladder), and other three-level quantum systems with one forbidden transition (referred to here as $\mathrm{\ensuremath{\Lambda}}$-type systems) play an important role in quantum physics. Various applications require manipulation of such systems using as a control a shaped laser field. In this work, we study how the degeneracy of energy states or of Bohr frequencies in these systems affects the efficiency or difficulty of finding the optimal shape of the control field. For this, we adopt the notion of higher-order traps, which was introduced in Pechen and Tannor, Phys. Rev. Lett. 106, 120402 (2011), where second-/third-order traps were discovered for a $\mathrm{\ensuremath{\Lambda}}$-atom with one forbidden transition and with nondegenerate energy levels. We theoretically study the control of such systems with and without degeneracy in their eigenstates and Bohr frequencies, and we investigate numerically using GRAPE and l-BFGS algorithms how this degeneracy influences the efficiency of optimizing the control laser field. We find that the degeneracy of the Bohr frequencies in the $\mathrm{\ensuremath{\Xi}}$ system, which makes the system energy levels symmetrically distributed, leads to the appearance of a seventh-order trap with a more significant attracting domain resulting in a more difficult optimization, while the degeneracy of energy states in generic $\mathrm{\ensuremath{\Lambda}}$-type systems does not lead to an increase of the order of the zero control trap compared to the nondegenerate case. We also find that when not only are the Bohr frequencies degenerate in the system $\mathrm{\ensuremath{\Xi}}$ but also the dipole moments for the two allowed transitions coincide (in this case, the $\mathrm{\ensuremath{\Xi}}$ system is not controllable), then true traps arise in the quantum control landscape. In particular, the constant zero control becomes a trap.
Abstract Last decades have faced considerable interest in various problems of quantum control. One of them is to determine which quantum systems are controllable, meaning that all possible states of such systems can be reached from any initial one under available controls. This paper focuses on controllability exploration of a specific four-level quantum system with three levels having the same energy and driven by an electromagnetic control field which allows transitions only between adjacent basis states. Such systems appear in the analysis of higher order traps in quantum control landscapes. This paper explores a few properties of the system, including a proof it is completely controllable for any complex values of the non-zero matrix elements of the interaction Hamiltonian.
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.