Quantum walks on graphs representing the firing patterns of a quantum neural network

Quantum walks have been shown to be fruitful tools in analysing the dynamic properties of quantum systems. This article proposes to use quantum walks as an approach to Quantum Neural Networks (QNNs). QNNs replace binary McCulloch-Pitts neurons with a qubit in order to use the advantages of quantum computing in neural networks. A quantum walk on the firing states of such a QNN is supposed to simulate central properties of the dynamics of classical neural networks, such as associative memory. It is shown that a biased discrete Hadamard walk derived from the updating process of a biological neuron does not lead to a unitary walk. However, a Stochastic Quantum Walk between the global firing states of a QNN can be constructed and it is shown that it contains the feature of associative memory. The quantum contribution to the walk accounts for a modest speed-up in some regimes.

[1]  Masoud Mohseni,et al.  Environment-assisted quantum transport , 2008, 0807.0929.

[2]  Julia Kempe,et al.  Quantum random walks: An introductory overview , 2003, quant-ph/0303081.

[3]  Andrzej Kossakowski,et al.  Properties of Quantum Markovian Master Equations , 1978 .

[4]  B. M. Fulk MATH , 1992 .

[5]  Ri-Gui Zhou,et al.  Quantum Associative Neural Network with Nonlinear Search Algorithm , 2012 .

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

[7]  C. Koch,et al.  Quantum mechanics in the brain , 2006, Nature.

[8]  Viv Kendon,et al.  Decoherence can be useful in quantum walks , 2002, quant-ph/0209005.

[9]  M. K. Ali,et al.  A Quantum Neural Network Model , 2002 .

[10]  S. Lloyd,et al.  Environment-assisted quantum walks in photosynthetic energy transfer. , 2008, The Journal of chemical physics.

[11]  Peter Dayan,et al.  Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems , 2001 .

[12]  Raúl Rojas,et al.  Neural Networks - A Systematic Introduction , 1996 .

[13]  P. Laguna,et al.  Signal Processing , 2002, Yearbook of Medical Informatics.

[14]  G. Lindblad On the generators of quantum dynamical semigroups , 1976 .

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

[16]  Barry C. Sanders,et al.  Quantum walks in higher dimensions , 2002 .

[17]  D. Signorini,et al.  Neural networks , 1995, The Lancet.

[18]  Sanjay Gupta,et al.  Quantum Neural Networks , 2001, J. Comput. Syst. Sci..

[19]  Max Tegmark,et al.  The importance of quantum decoherence in brain processes , 1999, ArXiv.

[20]  K. Birgitta Whaley,et al.  Quantum random-walk search algorithm , 2002, quant-ph/0210064.

[21]  Andris Ambainis,et al.  Quantum walks driven by many coins , 2002, quant-ph/0210161.

[22]  G. Fleming,et al.  Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature , 2009, Proceedings of the National Academy of Sciences.

[23]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[24]  廣瀬雄一,et al.  Neuroscience , 2019, Workplace Attachments.

[25]  Vivien M. Kendon,et al.  Decoherence in quantum walks – a review , 2006, Mathematical Structures in Computer Science.

[26]  James D. Whitfield,et al.  Quantum Stochastic Walks: A Generalization of Classical Random Walks and Quantum Walks , 2009, 0905.2942.

[27]  P. Ribeiro,et al.  Aperiodic quantum random walks. , 2004, Physical review letters.

[28]  T. Mančal,et al.  Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems , 2007, Nature.

[29]  Subhash Kak,et al.  Quantum Neural Computing , 1995 .

[30]  A. Selverston,et al.  Dynamical principles in neuroscience , 2006 .

[31]  Edward Farhi,et al.  An Example of the Difference Between Quantum and Classical Random Walks , 2002, Quantum Inf. Process..

[32]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[33]  Andris Ambainis,et al.  QUANTUM WALKS AND THEIR ALGORITHMIC APPLICATIONS , 2003, quant-ph/0403120.

[34]  Salvador Elías Venegas-Andraca,et al.  Quantum walks: a comprehensive review , 2012, Quantum Information Processing.

[35]  資訊工程學系 3/3_陳英瑞教授_department of computer science , 2011 .

[36]  K. B. Whaley,et al.  Limits of quantum speedup in photosynthetic light harvesting , 2009, 0910.1847.

[37]  Philip Ball,et al.  Physics of life: The dawn of quantum biology , 2011, Nature.

[38]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[39]  Justin R Caram,et al.  Long-lived quantum coherence in photosynthetic complexes at physiological temperature , 2010, Proceedings of the National Academy of Sciences.

[40]  E. Farhi,et al.  Quantum computation and decision trees , 1997, quant-ph/9706062.

[41]  Colin P. Williams,et al.  Quantum Neural Nets , 1998 .

[42]  G. J. Milburn,et al.  Implementing the quantum random walk , 2002 .

[43]  Andris Ambainis,et al.  Quantum random walks with decoherent coins , 2003 .

[44]  Denis Bernard,et al.  Open quantum random walks: Bistability on pure states and ballistically induced diffusion , 2013, 1303.6658.

[45]  Igor Jex,et al.  Recurrence of biased quantum walks on a line , 2009, 0902.3600.

[46]  W. Freeman,et al.  Dissipation and spontaneous symmetry breaking in brain dynamics , 2007, q-bio/0701053.

[47]  Frederick W. Strauch,et al.  Connecting the discrete- and continuous-time quantum walks , 2006 .

[48]  Richard W. Hamming,et al.  Error detecting and error correcting codes , 1950 .