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Tambov University Reports Series Natural and Technical Sciences

Критические точки целевых функционалов для задач управления квантовыми системами

Александр Николаевич Печень

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

In this talk we consider critical points for a wide class of objective functionals for controlled quantum systems. For closed quantum control systems, we show the existence of second order traps critical points which are not global maxima and where Hessian of the objective functional is negative semidefinite. For open quantum systems, the conditions for the absence of traps (maxima which are not global) are provided.

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In this talk we consider critical points for a wide class of objective functionals for controlled quantum systems. For closed quantum control systems, we show the existence of second order traps critical points which are not global maxima and where Hessian of the objective functional is negative semidefinite. For open quantum systems, the conditions for the absence of traps (maxima which are not global) are provided.
We study unconstrained control of a two-level quantum system and analyse critical points of the objective functional which represents quantum average of system observable at some final time TTT. In Proc. Steklov Inst. Math. 285, 233-240 (2014) it was shown that all maxima and minima of the objective functional are global if T≥πT\ge πT≥π (in suitable units). In the present work we show that all maxima and minima are global as soon as T≥π/2T\ge π/2T≥π/2. Hence we reduce by the factor of two the minimal time for which traps, i.e., local but not global maxima or minima of the objective functional, do not exist.
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 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.
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.