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The quest for a Quantum Neural Network

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Abstract

With the overwhelming success in the field of quantum information in the last decades, the ‘quest’ for a Quantum Neural Network (QNN) model began in order to combine quantum computing with the striking properties of neural computing. This article presents a systematic approach to QNN research, which so far consists of a conglomeration of ideas and proposals. Concentrating on Hopfield-type networks and the task of associative memory, it outlines the challenge of combining the nonlinear, dissipative dynamics of neural computing and the linear, unitary dynamics of quantum computing. It establishes requirements for a meaningful QNN and reviews existing literature against these requirements. It is found that none of the proposals for a potential QNN model fully exploits both the advantages of quantum physics and computing in neural networks. An outlook on possible ways forward is given, emphasizing the idea of Open Quantum Neural Networks based on dissipative quantum computing.

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Notes

  1. The only systematic review in the field of QNNs was given in 2000 by Ezhov and Ventura [22] (not counting the brief overview of the different types of implementations of QNNs in Oliveira et al. [19]). To our knowledge, there is no recent comprehensive review.

  2. The Hamming distance is the number of state flips to turn one binary string into another one, thus measuring the overlap between two binary strings [30].

  3. A problem is linearly separable if the respective outputs in phase space can be divided by a hyperplane. Perceptrons can only compute linear separable problems.

References

  1. Rabinovich, M.I., Varona, P., Selverston, A.I., Abarbanel, H.D.I.: Dynamical principles in neuroscience. Rev Mod. Phys. 78(4), 1213 (2006)

    Article  ADS  Google Scholar 

  2. McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biol. 5(4), 115–133 (1943)

    MathSciNet  MATH  Google Scholar 

  3. Kak, C.S.: Quantum neural computing. Adv. Imaging Electron Phys. 94, 259–313 (1995)

    Article  Google Scholar 

  4. Freeman, W.J., Vitiello, G.: Dissipation and spontaneous symmetry breaking in brain dynamics. J. Phys. A Math. Theor. 41(30), 304042 (2008)

    Article  MathSciNet  Google Scholar 

  5. Hameroff, S.: Quantum computation in brain microtubules? The penrose-hameroff ’orch or’ model of consciousness. Philos. Trans. Math. Phys. Eng. Sci. 356(1743), 1869–1896 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  6. Tegmark, M.: Importance of quantum decoherence in brain processes. Phys. Rev. E 61(4), 4194 (2000)

    Article  ADS  Google Scholar 

  7. Massimo Panella and Giuseppe Martinelli: Neural networks with quantum architecture and quantum learning. Int. J. Circuit Theory Appl. 39(1), 61–77 (2011)

    Article  Google Scholar 

  8. Gerasimos, G.: Neurodynamics and attractors in quantum associative memories. Integr. Comput. Aided Eng. 14(3), 225–242 (2007)

    Google Scholar 

  9. Lloyd, S., Mohseni, M., Rebentrost, P.: Quantum algorithms for supervised and unsupervised machine learning (preprint). arXiv:1307.0411 (2013)

  10. Aïmeur, E., Brassard, G., Gambs, S.: Quantum speed-up for unsupervised learning. Mach. Learn. 90(2), 261–287 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wiebe, Nathan., Kapoor, Ashish., Svore, Krysta.: Quantum nearest-neighbor algorithms for machine learning (preprint). arXiv:1401.2142 (2014)

  12. Shor, W.P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Andrecut, M., Ali, M.K.: A quantum neural network model. Int. J. Mod. Phys. C 13(01), 75–88 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Altaisky, M.V.: Quantum neural network (preprint). quant-ph/0107012 (2001)

  15. Sanjay Gupta and RKP Zia: Quantum neural networks. J. Comput. Syst. Sci. 63(3), 355–383 (2001)

    Article  Google Scholar 

  16. lisabeth, E., Behrman, C., Chandrashekar, V., Wang, Z., Belur, C.K., Steck, James, E., Skinner, S.R.: A quantum neural network computes entanglement (preprint) quant-ph/0202131 (2002)

  17. Li, F., Zheng, B.: A study of quantum neural networks. In: Proceedings of the 2003 International Conference on Neural Networks and Signal Processing, 2003, vol. 1, pp. 539–542. IEEE (2003)

  18. Zhou, R., Wang, H., Qian, W., Shi, Y.: Quantum associative neural network with nonlinear search algorithm. Int. J. Theor. Phys. 51(3), 705–723 (2012)

    Article  MATH  Google Scholar 

  19. Oliveira, W., Silva, A.J., Ludermir, T.B., Leonel, A., Galindo, W.R., Pereira, J.C.C.: Quantum logical neural networks. In: 10th Brazilian Symposium on Neural Networks, 2008. SBRN’08. pp. 147–152. IEEE (2008)

  20. Tóth, G., Lent, C.S., Tougaw, P.D., Brazhnik, Y., Weng, W., Porod, W., Liu, R.-W., Huang, Y.-F.: Quantum cellular neural networks (preprint). cond-mat/0005038 (2000)

  21. Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  22. Ezhov, A., Ventura, D.: Quantum neural networks. In: Future directions for intelligent systems and information sciences, pp. 213–234 (2000)

  23. Breuer, H.P., Petruccione, F.: The theory of open quantum systems. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  24. Verstraete, F., Wolf, M.M., Cirac, J.I.: Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5(9), 633–636 (2009)

    Article  Google Scholar 

  25. Purves, D.: Neuroscience, 3rd edn. Sinauer, Sunderland (2008)

    Google Scholar 

  26. Abbott, L.F., Regehr, W.G.: Synaptic computation. Nature 431(7010), 796–803 (2004)

    Article  ADS  Google Scholar 

  27. Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. 79(8), 2554–2558 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  28. Rojas, R.: Neural Nets: A Systematic Introduction. Springer, New York (1996)

    Google Scholar 

  29. John, J.: Hopfield and David W Tank. Computing with neural circuits- a model. Science 233(4764), 625–633 (1986)

    Article  Google Scholar 

  30. Richard, W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 29(2), 147–160 (1950)

    Article  Google Scholar 

  31. Amit, D.J., Gutfreund, H., Sompolinsky, H.: Spin-glass models of neural networks. Phys. Rev. A 32(2), 1007–1018 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  32. van Hemmen, J.L.: Spin-glass models of a neural network. Phys. Rev. A 34(4), 3435–3445 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  33. Hertz, J.A., Krogh, A.S., Palmer, R.G.: Introduction to the Theory of Neural Computation, vol. 1. Westview Press, Boulder (1991)

    Google Scholar 

  34. Mehrotra, K., Mohan, C.K., Ranka, S.: Elements of Artificial Neural Networks. MIT Press, Cambridge (1996)

    Google Scholar 

  35. Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. 81(10), 3088–3092 (1984)

    Article  ADS  Google Scholar 

  36. DiVincenzo, D.P. et al.: The physical implementation of quantum computation. (preprint) quant-ph/0002077 (2000)

  37. Briegel, H.J., Browne, D.E., Dür, W., Raussendorf, R., Van den Nest, M.: Measurement-based quantum computation. Nat. Phys. 5(1), 19–26 (2009)

    Article  Google Scholar 

  38. Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution (preprint). quant-ph/0001106 (2000)

  39. Gui Lu Long: Duality quantum computing and duality quantum information processing. Int. J. Theor. Phys. 50(4), 1305–1318 (2011)

    Article  MATH  Google Scholar 

  40. Menneer, T., Narayanan, A.: Quantum-inspired neural networks. Department of Computer Science, University of Exeter, UK, Technical Report 329 (1995)

  41. Peruš, M.: Neural networks as a basis for quantum associative networks. Neural Netw. World 10(6), 1001–1013 (2000)

    Google Scholar 

  42. Zak, M., Williams, C.P.: Quantum neural nets. Int. J. Theor. Phys. 37(2), 651–684 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Behrman, E.C., Steck, J.E., Skinner, S.R.: A spatial quantum neural computer. In: International Joint Conference on Neural Networks, 1999. IJCNN’99. vol. 2, pp. 874–877. IEEE (1999)

  44. Faber, J., Giraldi, G.A.: Quantum models for artificial neural networks. Electronically available: http://arquivosweb.lncc.br/pdfs/QNN-Review.pdf (2002)

  45. Silva, A.J., Ludermir, T.B., de Oliveira, W.R.: Superposition based learning algorithm. In: 2010 Eleventh Brazilian Symposium on Neural Networks (SBRN), pp. 1–6. IEEE (2010)

  46. Goncalves, C.P.: Quantum cybernetics and complex quantum systems science—a quantum connectionist exploration (preprint). arXiv:1402.1141v1 (2014)

  47. Ventura, D., Martinez, T.: Quantum associative memory. Inf. Sci. 124(1), 273–296 (2000)

    Article  MathSciNet  Google Scholar 

  48. Trugenberger, A.C.: Probabilistic quantum memories. Phys. Rev. Lett. 87, 067901 (Jul 2001)

  49. Andrecut, M., Ali, M.K.: Quantum associative memory. Int. J. Mod. Phys. B 17(12), 2447–2472 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Siomau, M.: Quantum perceptron (preprint). arXiv:1210.6626 (2012)

  51. Zak, M., Williams, C.P.: Quantum Recurrent Networks for Simulating Stochastic Processes. Springer, Berlin (1999)

    Book  Google Scholar 

  52. Behrman, E.C., Nash, L.R., Steck, J.E., Chandrashekar, A.A., Skinner, S.R.: Simulations of quantum neural networks. Inf. Sci. 128(3), 257–269 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  53. Behrman, E.C., Steck, J.E.: A quantum neural network computes its own relative phase (preprint). arXiv:1301.2808 (2013)

  54. da Adenilton, J., de Silva, W.R., Oliveira, T.B.Ludermir: Classical and superposed learning for quantum weightless neural networks. Neurocomputing 75(1), 52–60 (2012)

    Article  Google Scholar 

  55. Long, G.-L., Sun, Y.: Efficient scheme for initializing a quantum register with an arbitrary superposed state. Phys. Rev. A 64(1), 014303 (2001)

    Article  ADS  Google Scholar 

  56. Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching (preprint). quant-ph/9605034 (1996)

  57. Trugenberger, C.A.: Quantum pattern recognition. Quantum Inf. Process. 1(6), 471–493 (2002)

    Article  MathSciNet  Google Scholar 

  58. Sagheer, A., Zidan, M.: Autonomous quantum perceptron neural network (preprint). arXiv:1312.4149v1 (2013)

  59. Weigang, L.: Entangled neural networks. Department of Computer Science, University of Brasilia, CP, 4466, 70919-970 (2000)

  60. Segher, A.A., Metwally, N.: Communication via quantum neural network (preprint) arXiv:0912.2512 (2009)

  61. Neigovzen, R., Neves, J.L., Sollacher, R., Glaser, S.J.: Quantum pattern recognition with liquid-state nuclear magnetic resonance. Phys. Rev. A. 79(4), 042321 (2009)

    Article  ADS  Google Scholar 

  62. Li, F., Zhao, S., Zheng, B.: Quantum neural network in speech recognition. In: 6th International Conference on Signal Processing, 2002, vol. 2, pp. 1267–1270. IEEE (2002)

  63. Purushothaman, G., Karayiannis, N.B.: Quantum neural networks (qnns): inherently fuzzy feedforward neural networks. IEEE Trans. Neural Netw. 8(3), 679–693 (1997)

    Article  Google Scholar 

  64. Schneidman, E., Michael, J.B., Ronen, S., William, B.: Weak pairwise correlations imply strongly correlated network states in a neural population. Nature 440, 1007–1012 (2006)

    Article  ADS  Google Scholar 

  65. Nishimori, H., Nonomura, Y.: Quantum effects in neural networks. J. Phys. Soc. Jpn. 65(12), 3780–3796 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  66. Attal, S., Petruccione, F., Sinayskiy, I.: Open quantum walks on graphs. Phys. Lett. A 376(18), 1545–1548 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  67. Sinayskiy, I., Petruccione, F.: Efficiency of open quantum walk implementation of dissipative quantum computing algorithms. Quantum Inf. Process. 11(5), 1301–1309 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  68. Bauer, M., Bernard, D., Tilloy, A.: Open quantum random walks: bistability on pure states and ballistically induced diffusion. Phys. Rev. A 88, 062340 (Dec 2013)

  69. Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44(4), 307–327 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  70. Schuld, M., Sinayskiy, I., Petruccione, F.: Quantum walks on graphs representing the firing patterns of a quantum neural network. Phys. Rev. A 89(3), 032333 (2014)

    Article  ADS  Google Scholar 

  71. Ball, P.: The dawn of quantum biology. Nature 474(7351), 272–274 (2011)

    Article  ADS  Google Scholar 

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Acknowledgments

This work is based upon research supported by the South African Research Chair Initiative of the Department of Science and Technology and National Research Foundation.

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Authors and Affiliations

  1. Quantum Research Group, School of Chemistry and Physics, University of KwaZulu-Natal, Durban, KwaZulu-Natal, 4001, South Africa

    Maria Schuld, Ilya Sinayskiy & Francesco Petruccione

  2. National Institute for Theoretical Physics (NITheP), KwaZulu-Natal, 4001, South Africa

    Ilya Sinayskiy & Francesco Petruccione

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  1. Maria Schuld
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Correspondence to Maria Schuld.

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Schuld, M., Sinayskiy, I. & Petruccione, F. The quest for a Quantum Neural Network. Quantum Inf Process 13, 2567–2586 (2014). https://doi.org/10.1007/s11128-014-0809-8

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