Quantum learning of classical stochastic processes

Among several tasks in Machine Learning, is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of such problem is the task of inferring the Hidden Markov Model underlying a given stochastic process. This is known a...

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Autores: Monras Blasi, Àlex, Winter, Andreas|||0000-0001-6344-4870
Tipo de recurso: capítulo de libro
Fecha de publicación:2014
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:204038
Acceso en línea:https://ddd.uab.cat/record/204038
https://dx.doi.org/urn:doi:10.4230/LIPIcs.TQC.2014.99
Access Level:acceso abierto
Palabra clave:Quantum instrument
Hidden Markov model
Machine learning
Quantum measurement
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spelling Quantum learning of classical stochastic processesThe completely-positive realization problemMonras Blasi, ÀlexWinter, Andreas|||0000-0001-6344-4870Quantum instrumentHidden Markov modelMachine learningQuantum measurementAmong several tasks in Machine Learning, is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of such problem is the task of inferring the Hidden Markov Model underlying a given stochastic process. This is known as the positive realization problem (PRP) [3] and constitutes a central problem in machine learning. The PRP and its solutions have far-reaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory [21]. We consider the scenario where the latent variables are quantum (e. g., quantum states of a finite-dimensional system), and the system dynamics is constrained only by physical transformations on the quantum system. The observable dynamics is then described by a quantum instrument, and the task is to determine which quantum instrument - if any - yields the process at hand by iterative application. We take as a starting point the theory of quasi-realizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors. This description, despite its remarkable resemblance with the Hidden Markov Model, or the iterated quantum instrument, is however devoid from any stochastic or quantum mechanical interpretation, as said maps fail to satisfy any positivity conditions. The Completely-Positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists. We generalize some key results of stochastic realization theory, and show that the problem has deep connections with operator systems theory, giving possible insight to the lifting problem in quotient operator systems. Our results have potential applications in quantum machine learning, device-independent characterization and reverse-engineering of stochastic processes and quantum processors, and more generally, of dynamical processes with quantum memory [16, 17].Leibniz-Zentrum fuer Informatik 22014-01-0120142014-01-01Capítol de llibrehttp://purl.org/coar/resource_type/c_3248VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/bookPartapplication/pdfhttps://ddd.uab.cat/record/204038https://dx.doi.org/urn:doi:10.4230/LIPIcs.TQC.2014.99reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Ciencia e Innovación https://doi.org/10.13039/501100004837 FIS2008-01236open accesshttp://purl.org/coar/access_right/c_abf2Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2040382026-06-06T12:50:31Z
dc.title.none.fl_str_mv Quantum learning of classical stochastic processes
The completely-positive realization problem
title Quantum learning of classical stochastic processes
spellingShingle Quantum learning of classical stochastic processes
Monras Blasi, Àlex
Quantum instrument
Hidden Markov model
Machine learning
Quantum measurement
title_short Quantum learning of classical stochastic processes
title_full Quantum learning of classical stochastic processes
title_fullStr Quantum learning of classical stochastic processes
title_full_unstemmed Quantum learning of classical stochastic processes
title_sort Quantum learning of classical stochastic processes
dc.creator.none.fl_str_mv Monras Blasi, Àlex
Winter, Andreas|||0000-0001-6344-4870
author Monras Blasi, Àlex
author_facet Monras Blasi, Àlex
Winter, Andreas|||0000-0001-6344-4870
author_role author
author2 Winter, Andreas|||0000-0001-6344-4870
author2_role author
dc.subject.none.fl_str_mv Quantum instrument
Hidden Markov model
Machine learning
Quantum measurement
topic Quantum instrument
Hidden Markov model
Machine learning
Quantum measurement
description Among several tasks in Machine Learning, is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of such problem is the task of inferring the Hidden Markov Model underlying a given stochastic process. This is known as the positive realization problem (PRP) [3] and constitutes a central problem in machine learning. The PRP and its solutions have far-reaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory [21]. We consider the scenario where the latent variables are quantum (e. g., quantum states of a finite-dimensional system), and the system dynamics is constrained only by physical transformations on the quantum system. The observable dynamics is then described by a quantum instrument, and the task is to determine which quantum instrument - if any - yields the process at hand by iterative application. We take as a starting point the theory of quasi-realizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors. This description, despite its remarkable resemblance with the Hidden Markov Model, or the iterated quantum instrument, is however devoid from any stochastic or quantum mechanical interpretation, as said maps fail to satisfy any positivity conditions. The Completely-Positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists. We generalize some key results of stochastic realization theory, and show that the problem has deep connections with operator systems theory, giving possible insight to the lifting problem in quotient operator systems. Our results have potential applications in quantum machine learning, device-independent characterization and reverse-engineering of stochastic processes and quantum processors, and more generally, of dynamical processes with quantum memory [16, 17].
publishDate 2014
dc.date.none.fl_str_mv 2
2014-01-01
2014
2014-01-01
dc.type.none.fl_str_mv Capítol de llibre
http://purl.org/coar/resource_type/c_3248
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/bookPart
format bookPart
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/204038
https://dx.doi.org/urn:doi:10.4230/LIPIcs.TQC.2014.99
url https://ddd.uab.cat/record/204038
https://dx.doi.org/urn:doi:10.4230/LIPIcs.TQC.2014.99
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 FIS2008-01236
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Leibniz-Zentrum fuer Informatik
publisher.none.fl_str_mv Leibniz-Zentrum fuer Informatik
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
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