Uncomputability and complexity of quantum control

Abstract In laboratory and numerical experiments, physical quantities are known with a finite precision and described by rational numbers. Based on this, we deduce that quantum control problems both for open and closed systems are in general not algorithmically solvable, i.e., there is no algorithm...

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Autores principales:	Denys I. Bondar, Alexander N. Pechen
Formato:	article
Lenguaje:	EN
Publicado:	Nature Portfolio 2020
Materias:	Medicine R Science Q
Acceso en línea:	https://doaj.org/article/4cb6e27a82bc454785a683b3dd175e6b
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spelling	oai:doaj.org-article:4cb6e27a82bc454785a683b3dd175e6b2021-12-02T14:16:42ZUncomputability and complexity of quantum control10.1038/s41598-019-56804-12045-2322https://doaj.org/article/4cb6e27a82bc454785a683b3dd175e6b2020-01-01T00:00:00Zhttps://doi.org/10.1038/s41598-019-56804-1https://doaj.org/toc/2045-2322Abstract In laboratory and numerical experiments, physical quantities are known with a finite precision and described by rational numbers. Based on this, we deduce that quantum control problems both for open and closed systems are in general not algorithmically solvable, i.e., there is no algorithm that can decide whether dynamics of an arbitrary quantum system can be manipulated by accessible external interactions (coherent or dissipative) such that a chosen target reaches a desired value. This conclusion holds even for the relaxed requirement of the target only approximately attaining the desired value. These findings do not preclude an algorithmic solvability for a particular class of quantum control problems. Moreover, any quantum control problem can be made algorithmically solvable if the set of accessible interactions (i.e., controls) is rich enough. To arrive at these results, we develop a technique based on establishing the equivalence between quantum control problems and Diophantine equations, which are polynomial equations with integer coefficients and integer unknowns. In addition to proving uncomputability, this technique allows to construct quantum control problems belonging to different complexity classes. In particular, an example of the control problem involving a two-mode coherent field is shown to be NP-hard, contradicting a widely held believe that two-body problems are easy.Denys I. BondarAlexander N. PechenNature PortfolioarticleMedicineRScienceQENScientific Reports, Vol 10, Iss 1, Pp 1-10 (2020)
institution	DOAJ
collection	DOAJ
language	EN
topic	Medicine R Science Q
spellingShingle	Medicine R Science Q Denys I. Bondar Alexander N. Pechen Uncomputability and complexity of quantum control
description	Abstract In laboratory and numerical experiments, physical quantities are known with a finite precision and described by rational numbers. Based on this, we deduce that quantum control problems both for open and closed systems are in general not algorithmically solvable, i.e., there is no algorithm that can decide whether dynamics of an arbitrary quantum system can be manipulated by accessible external interactions (coherent or dissipative) such that a chosen target reaches a desired value. This conclusion holds even for the relaxed requirement of the target only approximately attaining the desired value. These findings do not preclude an algorithmic solvability for a particular class of quantum control problems. Moreover, any quantum control problem can be made algorithmically solvable if the set of accessible interactions (i.e., controls) is rich enough. To arrive at these results, we develop a technique based on establishing the equivalence between quantum control problems and Diophantine equations, which are polynomial equations with integer coefficients and integer unknowns. In addition to proving uncomputability, this technique allows to construct quantum control problems belonging to different complexity classes. In particular, an example of the control problem involving a two-mode coherent field is shown to be NP-hard, contradicting a widely held believe that two-body problems are easy.
format	article
author	Denys I. Bondar Alexander N. Pechen
author_facet	Denys I. Bondar Alexander N. Pechen
author_sort	Denys I. Bondar
title	Uncomputability and complexity of quantum control
title_short	Uncomputability and complexity of quantum control
title_full	Uncomputability and complexity of quantum control
title_fullStr	Uncomputability and complexity of quantum control
title_full_unstemmed	Uncomputability and complexity of quantum control
title_sort	uncomputability and complexity of quantum control
publisher	Nature Portfolio
publishDate	2020
url	https://doaj.org/article/4cb6e27a82bc454785a683b3dd175e6b
work_keys_str_mv	AT denysibondar uncomputabilityandcomplexityofquantumcontrol AT alexandernpechen uncomputabilityandcomplexityofquantumcontrol
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Uncomputability and complexity of quantum control

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