Quantum learning of classical stochastic processes: The completely positive realization problem
Abstract
Among several tasks in Machine Learning, a specially important one is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of this is the task of inferring the hidden Markov model underlying a given stochastic process. This is known as the positive realization problem (PRP), [L. Benvenuti and L. Farina, IEEE Trans. Autom. Control 49(5), 651–664 (2004)] and constitutes a central problem in machine learning. The PRP and its solutions have farreaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory. We consider the scenario where the latent variables are quantum (i.e., quantum states of a finitedimensional 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 quasirealizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilitiesmore »
 Authors:

 Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona) (Spain)
 Publication Date:
 OSTI Identifier:
 22479622
 Resource Type:
 Journal Article
 Journal Name:
 Journal of Mathematical Physics
 Additional Journal Information:
 Journal Volume: 57; Journal Issue: 1; Other Information: (c) 2016 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 00222488
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONTROL THEORY; FUNCTIONALS; ITERATIVE METHODS; LEARNING; MAPS; MARKOV PROCESS; MATHEMATICAL SOLUTIONS; QUANTUM MECHANICS; QUANTUM STATES; QUANTUM SYSTEMS; VECTORS
Citation Formats
Monràs, Alex, Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Winter, Andreas, Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, and ICREA—Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys, 23, 08010 Barcelona. Quantum learning of classical stochastic processes: The completely positive realization problem. United States: N. p., 2016.
Web. doi:10.1063/1.4936935.
Monràs, Alex, Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Winter, Andreas, Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, & ICREA—Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys, 23, 08010 Barcelona. Quantum learning of classical stochastic processes: The completely positive realization problem. United States. doi:10.1063/1.4936935.
Monràs, Alex, Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Winter, Andreas, Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, and ICREA—Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys, 23, 08010 Barcelona. Fri .
"Quantum learning of classical stochastic processes: The completely positive realization problem". United States. doi:10.1063/1.4936935.
@article{osti_22479622,
title = {Quantum learning of classical stochastic processes: The completely positive realization problem},
author = {Monràs, Alex and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 and Winter, Andreas and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 and ICREA—Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys, 23, 08010 Barcelona},
abstractNote = {Among several tasks in Machine Learning, a specially important one is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of this is the task of inferring the hidden Markov model underlying a given stochastic process. This is known as the positive realization problem (PRP), [L. Benvenuti and L. Farina, IEEE Trans. Autom. Control 49(5), 651–664 (2004)] and constitutes a central problem in machine learning. The PRP and its solutions have farreaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory. We consider the scenario where the latent variables are quantum (i.e., quantum states of a finitedimensional 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 quasirealizations, 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 of 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, deviceindependent characterization and reverseengineering of stochastic processes and quantum processors, and more generally, of dynamical processes with quantum memory [M. Guţă, Phys. Rev. A 83(6), 062324 (2011); M. Guţă and N. Yamamoto, eprint http://arxiv.org/abs/1303.3771 (2013)].},
doi = {10.1063/1.4936935},
journal = {Journal of Mathematical Physics},
issn = {00222488},
number = 1,
volume = 57,
place = {United States},
year = {2016},
month = {1}
}