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Proceedings of the Steklov Institute of Mathematics

Conditions for the Absence of Local Extrema in Problems of Quantum Coherent Control

Nikolay Borisovich Ilin, Alexander Nikolaevich Pechen

Published May 1, 2018• 4 citations
Publisher

Abstract

We consider a terminal control problem for quantum systems which is formulated as the problem of maximizing the objective functional at some fixed finite time. Within the framework of this problem, we discuss known results on the local maxima of the objective functional that are not global. This question is important for quantum control, since such local maxima could make it difficult to find the global maximum by local search in numerical optimization or under laboratory conditions.

DOI
10.1134/s0081543818040090

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We consider a terminal control problem for quantum systems which is formulated as the problem of maximizing the objective functional at some fixed finite time. Within the framework of this problem, we discuss known results on the local maxima of the objective functional that are not global. This question is important for quantum control, since such local maxima could make it difficult to find the global maximum by local search in numerical optimization or under laboratory conditions.
Many phenomena in physics, chemistry, and biology involve seeking an optimal control to maximize an objective for a classical or quantum system which is open and interacting with its environment. The complexity of finding an optimal control for maximizing an objective is strongly affected by the possible existence of sub-optimal maxima. Within a unified framework under specified conditions, control objectives for maximizing at a terminal-time physical observables of open classical and quantum systems are shown to be inherently free of sub-optimal maxima. This attractive feature is of central importance for enabling the discovery of controls in a seamless fashion in a wide range of phenomena transcending the quantum and classical regimes.
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.
We consider the Mayer maximization problem for an objective functional that describes the average value at some fixed time of a quantum-mechanical observable for a two-level quantum system (qubit). In the previous studies we proved that for sufficiently large times the objective functional has no local maxima that are not global maxima. Such local maxima that are not global are called traps. In this paper we prove that for sufficiently short times under certain conditions traps for this problem do exist.
In the present work the extrema of the objective functional for the problem of generation of quantum gates (logical elements for quantum computations) for two--level systems are investigated for short duration of the control. The problem of existence of local but not global extrema, the so called traps, is considered. In prior works the absence of traps was proved for a sufficiently long control duration. In this paper we prove that for almost all target unitary operators and system Hamiltonians traps are absent for an arbitrarily small control duration. For the remainder target unitary operators and Hamiltonians we obtain a new estimate for the lower boundary of the control duration which guarantees the absence of traps.
A broad class of quantum control problems entails optimizing the expectation value of an observable operator through tailored unitary propagation of the system density matrix. Such optimization processes can be viewed as a directed search over a quantum control landscape. The attainment of the global extrema of this landscape is the goal of quantum control. Local optima will generally exist, and their enumeration is shown to scale factorially with the system's effective Hilbert space dimension. A Hessian analysis reveals that these local optima have saddlepoint topology and cannot behave as suboptimal extrema traps. The implications of the landscape topology for practical quantum control efforts are discussed, including in the context of nonideal operating conditions.