Quantum hidden Markov models based on transition operation matrices

In this work, we extend the idea of quantum Markov chains (Gudder in J Math Phys 49(7):072105 [3]) in order to propose quantum hidden Markov models (QHMMs). For that, we use the notions of transition operation matrices and vector states, which are an extension of classical stochastic matrices and probability distributions. Our main result is the Mealy QHMM formulation and proofs of algorithms needed for application of this model: Forward for general case and Vitterbi for a restricted class of QHMMs. We show the relations of the proposed model to other quantum HMM propositions and present an example of application.

[1]  F. Petruccione,et al.  Open Quantum Walks on Graphs , 2012, 1401.3305.

[2]  Bart De Moor,et al.  Equivalence of state representations for hidden Markov models , 2007, 2007 European Control Conference (ECC).

[3]  Munther A. Dahleh,et al.  Minimal realization problem for Hidden Markov Models , 2014, 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[4]  Zoubin Ghahramani,et al.  An Introduction to Hidden Markov Models and Bayesian Networks , 2001, Int. J. Pattern Recognit. Artif. Intell..

[5]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[6]  Norio Konno,et al.  Limit Theorems for Open Quantum Random Walks , 2012, 1209.1419.

[7]  Piotr Gawron,et al.  Generalized Open Quantum Walks on Apollonian Networks , 2014, PloS one.

[8]  F. Petruccione,et al.  Properties of open quantum walks on $\mathbb {Z}$ , 2012 .

[9]  Stanley Gudder,et al.  Quantum Markov chains , 2008 .

[10]  J. Billingsley.  Mathematics for Control , 2005 .

[11]  K. Wiesner,et al.  Hidden Quantum Markov Models and non-adaptive read-out of many-body states , 2010, 1002.2337.

[12]  Mathukumalli Vidyasagar,et al.  The complete realization problem for hidden Markov models: a survey and some new results , 2011, Math. Control. Signals Syst..

[13]  F. Petruccione,et al.  Open Quantum Random Walks , 2012, 1402.3253.

[14]  F. Petruccione,et al.  Open Quantum Walks: a short introduction , 2013, 1402.2146.

[15]  Francesco Petruccione,et al.  Efficiency of open quantum walk implementation of dissipative quantum computing algorithms , 2012, Quantum Inf. Process..

[16]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[17]  G. Goldhaber.  On communication. , 1979, Hospital supervisor's bulletin.

[18]  Yuan Feng,et al.  Quantum Markov chains: Description of hybrid systems, decidability of equivalence, and model checking linear-time properties , 2015, Inf. Comput..

[19]  Clement Ampadu.  Averaging in SU（2） open quantum random walk , 2014 .

[20]  Lukasz Pawela,et al.  Central limit theorem for reducible and irreducible open quantum walks , 2014, Quantum Inf. Process..

[21]  Germany,et al.  Subnormalized states and trace-nonincreasing maps , 2007 .

[22]  Francesco Petruccione,et al.  Open Quantum Random Walks and the Open Quantum Pascal Triangle , 2011 .