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 Tr. Mat. Inst. Steklova, 2014, Volume 285, Pages 244–252 (Mi tm3552)

Coherent control of a qubit is trap-free

A. N. Pechen, N. B. Il'in

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: There is a strong interest in optimal manipulating of quantum systems by external controls. Traps are controls which are optimal only locally but not globally. If they exist, they can be serious obstacles to the search of globally optimal controls in numerical and laboratory experiments, and for this reason the analysis of traps attracts considerable attention. In this paper we prove that for a wide range of control problems for two-level quantum systems all locally optimal controls are also globally optimal. Hence we conclude that two-level systems in general are trap-free. In particular, manipulating qubits – two-level quantum systems forming a basic building block for quantum computation – is free of traps for fundamental problems such as the state preparation and gate generation.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-31115 This work was supported by the Russian Foundation for Basic Research, project no. 14-01-31115.

DOI: https://doi.org/10.1134/S0371968514020162

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English version:
Proceedings of the Steklov Institute of Mathematics, 2014, 285, 233–240

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Document Type: Article
UDC: 517.977.5

Citation: A. N. Pechen, N. B. Il'in, “Coherent control of a qubit is trap-free”, Selected topics of mathematical physics and analysis, Collected papers. In commemoration of the 90th anniversary of Academician Vasilii Sergeevich Vladimirov's birth, Tr. Mat. Inst. Steklova, 285, MAIK Nauka/Interperiodica, Moscow, 2014, 244–252; Proc. Steklov Inst. Math., 285 (2014), 233–240

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3552
• https://doi.org/10.1134/S0371968514020162
• http://mi.mathnet.ru/eng/tm/v285/p244

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This publication is cited in the following articles:
1. A. N. Pechen, N. B. Il'in, “Existence of traps in the problem of maximizing quantum observable averages for a qubit at short times”, Proc. Steklov Inst. Math., 289 (2015), 213–220
2. A. N. Pechen, N. B. Il'in, “On critical points of the objective functional for maximization of qubit observables”, Russian Math. Surveys, 70:4 (2015), 782–784
3. A. N. Pechen, “On the speed gradient method for generating unitary quantum operations for closed quantum systems”, Russian Math. Surveys, 71:3 (2016), 597–599
4. Alexander N. Pechen, Nikolay B. Il'in, “On the problem of maximizing the transition probability in an $n$-level quantum system using nonselective measurements”, Proc. Steklov Inst. Math., 294 (2016), 233–240
5. A. N. Pechen, N. B. Il'in, “On extrema of the objective functional for short-time generation of single-qubit quantum gates”, Izv. Math., 80:6 (2016), 1200–1212
6. A. N. Pechen, N. B. Il'in, “Control landscape for ultrafast manipulation by a qubit”, J. Phys. A-Math. Theor., 50:7 (2017), 075301
7. N. B. Il'in, A. N. Pechen, “Conditions for the absence of local extrema in problems of quantum coherent control”, Proc. Steklov Inst. Math., 301 (2018), 109–113
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