Critical Points of the Optimal Quantum Control Landscape: A Propagator Approach

The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in L2(0, T), however, there are critical points for which the fidelity can assume any value in (0, 1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in L2(0, T) is larger and includes traps, some of which remain traps even when the target time is allowed to vary.

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Experimental exploration over a quantum control landscape through nuclear magnetic resonance

Physical Review A ◽

10.1103/physreva.89.033413 ◽

2014 ◽

Vol 89 (3) ◽

Cited By ~ 16

Author(s):

Qiuyang Sun ◽

István Pelczer ◽

Gregory Riviello ◽

Re-Bing Wu ◽

Herschel Rabitz

Keyword(s):

Nuclear Magnetic Resonance ◽

Magnetic Resonance ◽

Quantum Control ◽

Control Landscape ◽

Nuclear Magnetic

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Common foundations of optimal control across the sciences: evidence of a free lunch

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2016.0210 ◽

2017 ◽

Vol 375 (2088) ◽

pp. 20160210 ◽

Cited By ~ 7

Author(s):

Benjamin Russell ◽

Herschel Rabitz

Keyword(s):

Evolutionary Biology ◽

Quantum Control ◽

Mathematical Framework ◽

Optimal Controls ◽

Control Variables ◽

Common Control ◽

Algorithmic Search ◽

Set Up ◽

Open Question ◽

Control Landscape

A common goal in the sciences is optimization of an objective function by selecting control variables such that a desired outcome is achieved. This scenario can be expressed in terms of a control landscape of an objective considered as a function of the control variables. At the most basic level, it is known that the vast majority of quantum control landscapes possess no traps, whose presence would hinder reaching the objective. This paper reviews and extends the quantum control landscape assessment, presenting evidence that the same highly favourable landscape features exist in many other domains of science. The implications of this broader evidence are discussed. Specifically, control landscape examples from quantum mechanics, chemistry and evolutionary biology are presented. Despite the obvious differences, commonalities between these areas are highlighted within a unified mathematical framework. This mathematical framework is driven by the wide-ranging experimental evidence on the ease of finding optimal controls (in terms of the required algorithmic search effort beyond the laboratory set-up overhead). The full scope and implications of this observed common control behaviour pose an open question for assessment in further work. This article is part of the themed issue ‘Horizons of cybernetical physics’.

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