Hidden Quantum Markov Models and Open Quantum Systems with Instantaneous Feedback

  • Lewis A. Clark
  • Wei Huang
  • Thomas M. Barlow
  • Almut Beige
Part of the Emergence, Complexity and Computation book series (ECC, volume 14)


Hidden Markov Models are widely used in classical computer science to model stochastic processes with a wide range of applications. This paper concerns the quantum analogues of these machines — so-called Hidden Quantum Markov Models (HQMMs). Using the properties of Quantum Physics, HQMMs are able to generate more complex random output sequences than their classical counterparts, even when using the same number of internal states. They are therefore expected to find applications as quantum simulators of stochastic processes. Here, we emphasise that open quantum systems with instantaneous feedback are examples of HQMMs, thereby identifying a novel application of quantum feedback control.


Stochastic Processes Hidden Markov Models Quantum Simulations Quantum Feedback 


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  1. 1.
    Norris, J.R.: Markov chains, Cambridge University Press (1998)Google Scholar
  2. 2.
    Rabiner, L.R.: A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77, 257 (1989)CrossRefGoogle Scholar
  3. 3.
    Xue, H.: Hidden Markov Models Combining Discrete Symbols and Continuous Attributes in Handwriting Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 28, 458 (2006)CrossRefGoogle Scholar
  4. 4.
    Vanluyten, B., Willems, J.C., Moor, B.D.: Equivalence of State Representations for Hidden Markov Models. Systems and Control Letters 57, 410 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Wiesner, K., Crutchfield, C.P.: Computation in finitary stochastic and quantum processes. Physica D 237, 1173 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Monras, A., Beige, A., Wiesner, K.: Hidden Quantum Markov Models and non-adaptive read-out of many-body states. App. Math. and Comp. Sciences 3, 93 (2011)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Gmeiner, P.: Equality conditions for internal entropies of certain classical and quantum models, arXiv:1108.5303 (2011)Google Scholar
  8. 8.
    O‘Neill, B., Barlow, T.M., Safranek, D., Beige, A.: Hidden Quantum Markov Models with one qubit. In: AIP Conf. Proc., vol. 1479, p. 667 (2012)Google Scholar
  9. 9.
    Sweke, R., Sinayskiy, I., Petruccione, F.: Simulation of Single-Qubit Open Quantum Systems, arXiv:1405.6049 (2014)Google Scholar
  10. 10.
    Kraus, K.: States, Effects and Operations. Lecture Notes in Physics, vol. 190. Springer, Berlin (1983)zbMATHGoogle Scholar
  11. 11.
    Wiseman, H.M., Milburn, G.J.: Quantum Measurement and Control. Cambridge University Press (2010)Google Scholar
  12. 12.
    Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Dalibard, J., Castin, Y., Molmer, K.: Wave-function approach to dissipative processes in quantum optics. Phys. Rev. Lett. 68, 580 (1992)CrossRefGoogle Scholar
  14. 14.
    Hegerfeldt, G.C.: How to reset an atom after a photon detection. Applications to photon counting processes. Phys. Rev. A 47, 449 (1993)CrossRefGoogle Scholar
  15. 15.
    Carmichael, H.: An Open Systems Approach to Quantum Optics. Lecture Notes in Physics, vol. 18. Springer, Berlin (1993)zbMATHGoogle Scholar
  16. 16.
    Goldenfeld, N., Woese, C.: Life is Physics: evolution as a collective phenomenon far from equilibrium. Ann. Rev. Cond. Matt. Phys. 2, 375 (2011)CrossRefGoogle Scholar
  17. 17.
    Schuld, M., Sinayskiy, I., Petruccione, F.: Quantum walks on graphs representing the firing patterns of a quantum neural network, arXiv:1404.0159 (2014)Google Scholar
  18. 18.
    Kiesslich, G., Emary, C., Schaller, G., Brandes, T.: Reverse quantum state engineering using electronic feedback loops. New. J. Phys. 14, 123036 (2012)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Emary, C.: Delayed feedback control in quantum transport. Phil. Trans. R. Soc. A 371, 1999 (2013)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Stokes, A., Kurcz, A., Spiller, T.P., Beige, A.: Extending the validity range of quantum optical master equations. Phys. Rev. A 85, 053805 (2012)Google Scholar
  21. 21.
    Lindblad, G.: On the generators of quantum dynamical semigroups. Comm. Math. Phys. 48, 119 (1976)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Lewis A. Clark
    • 1
  • Wei Huang
    • 2
  • Thomas M. Barlow
    • 1
  • Almut Beige
    • 1
  1. 1.The School of Physics and AstronomyUniversity of LeedsLeedsUK
  2. 2.20 Dover Drive SingaporeSingapore University of Technology & DesignSingaporeSingapore

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