Quantum Neural Computing Study

          This document is prepared for describing the basic idea of Quantum Neural Computing. It has been modified since the first version and will be better with new coming information. Please tell me about your research and opinion.
                                             Li Weigang, Brasilia, 1999 - 2003.

1. Quantum Computing
2. Quantum Computer Simulation
3. Problems of Artificial Neural Networks
4. Quantum Computing and Artificial Neural Networks
5. Entangled Neural Networks
6. Conferences
7. Acknowledgments
8. References

1. Quantum Computing

Since Beniof [Ben82] and Feynman [Fey82] discovered the possibility of using quantum mechanical system for reasonable computing and Deutsch [Deu85] defined the first quantum computing model, the quantum computation have been developed as a interesting multidiscipline. Specially in recent years, the appearances of Shor's factoring algorithm [Sho94] and Grover's search algorithm [Grov96] speeded up the development in this area. A general review of the area is illustrated by Ekert and Deutsch´s article [Ekert98].

Sequential processing: the conventional learning is in terms of serial processing and this slowed down the acceptance of a sampling operation that could achieve task-dependent selectivity in a parallel processing environment [Gros98].

Memoryless learning: humans do remember and recall information that is provided to them as a part of learning. For ANNs, training examples are not stored, rather they are used for the changing of the weights and immediately forgotten. (ICNN´97, Menneer and Narayanan, 1998).

Repetition of training examples: humans do not undergo the prolonged and repeated exposure to information that ANNs require to learn training patterns.

Catastrophic forgetting is the inability of a neural network to retain old information in the presence of new. (French 1991). (ICNN´97, Menneer and Narayanan, 1998).

In the field of artificial neural networks (ANN), some pioneers introduced quantum computation into analogous discussion such as quantum associative memory, parallel learning and empirical analysis [Chr95, Kak95, Men95, Zar95, Beh96, Pru96, Val96]. They constructed the foundation for further study of quantum computation in artificial neural networks. Menneer developed her Ph. D. thesis in which deeply discussed the application of quantum theory to ANN and demonstrated that QUANNs are more efficient than ANN for classification tasks [Menn98]. Ventura and Martinez developed a quantum associative memory, its structure and learning manner in quantum version [Ven98]. Weigang tried to develop a parallel-SOM and mentioned the once learning manner in quantum computing environment [Wei98]. He also proposed an Entanglement Neural Networks (ENN) on the base of the quantum teleportation and its extension with intelligent sense. In AI and quantum computing, there is more recent review from Hirsh et al. [Hir99, Hog99, Kak99, Ven99]. The followings are main points of their researches.

As opposed to the gradient descent paradigms based on connectionist concepts, the quantum holographic model indicates an extremely large capacity for learning and expression of complex stimulus-response associations within individual neuron cells in one non-iterative transformation (Schempp, 1993).

Quantum wave function collapse "is very similar toneuronal-pattern- reconstruction from memory. In memory, there is a superposition of many stored patterns. One of them is selectively ‘brought forward from the background’ if an external stimulus triggers such a reconstruction" (Perus 1995). After that, he also developed some interesting papers such as Mind: neural computing plus quantum consciousness (Perus, 1997) and Common mathematical foundations of neural and quantum informatics (Perus, 1998).

If our interpretation of quantum theory forces us to replace the classical principles of identity and reparability by some qualitatively new, holistic notions, and if quantum-sized processes of the brain cannot be ignored in the description of human cognition and experience, then we may have to re-evaluate the prospects of connectionism as a general framework for the description of human cognition and experience (Pylkko, 1995).

A quantum neural computer consists of an ANN where quantum processes are supported. The ANN is a self-organizing type that becomes a different measuring system based on associations triggered by an external or internally generated stimulus. [Kak95].

A quantum learning system might acquire some form of intentionally, and begin the bridging of the physical/mental gap (Christely, 1995).

Menneer and Narayanan (1995) have applied the multiple universes view from quantum theory to one-layer ANNs.

Behrman et al. (1996) open their description of a quantum dot neural network: Potentially, a quantum neural network would be an extremely powerful computational tool..., at least in principle, of performing computations that cannot be done, classically... an actual working quantumneural net would likey want to take advantage of the greater multiplicity and connectivity inherent in an entire array of quantum dot molecules, by placing molecules ohysically close enough to each other that nearest neighbors can interact directly ...

In 1997, Lagaris, Likas and Fotiadis (Lagaris et al, 1997) developed Artificial Neural Networks for Solving Ordinary and Partial Differential Equations.

The using of quantum parallelism is often connected with consideration of quantum system with huge dimension of space of states. The applications described further are used some other properties of quantum systems and they do not demand such huge number of states. The term "images recognition" is used here for some broad class of problems. (Vlasov, 1997).

Purushothaman and Karayiannis (1998) publicated their paper "quantum neural networks (qnns) - inherently fuzzy feedforward neural networks"   in IEEE Transactions on Neural Networks. It is also implimented in a classical ANN simulator - QNN/SNNS.

Quantum computation uses microscopic quantum level effects to perform computational tasks and has produced results that in some cases are exponentially faster than their classical counterparts. The unique characteristics of quantum theory may also be used to create a quantum associative memory with a capacity exponential in the number of neurons (Ventura and Martinez, 1998).

Quantum Artificial Neural Networks (QUANNs) are more efficient than Classical Artificial Neural Networks (CLANNs) for classification tasks, in that the time required for training is much less for QUANNs.(Menneer and Narayanan, 1998). The repetition is not required by QUANNs, since each component network learns only one pattern causing the training set to be learned more quickly. (Menneer and Narayanan, 1998).

In a Parallel Self-Organizing Map: the once learning manner is more similar to human learning and memorizing activities. During training, weight updating is managed through a sequence of operations among some transformation and operation matrices (Weigang, 1998).

Bieberich (Bieberich, 1999) developed a paper about "Non-local quantum evolution of entangled ensemble states in neural nets and its significance for brain function and a theory of consciousness" .

Quantum dissipation and neural net dynamics are discussed by Pessa and Vitiello [1999].

The recent rigorous proof of a quantum version of the Shannon- McMillan Theorem provides the foundation for analogous treatment of pure quantum neural networks, whose inherent parallelism should permit greatly enhanced pattern recognition. The reduction of a general quantum neural network to a parametized information source characterized by grammar and syntax implies the necessity of a quantum linguistics [Wallace, 2000].

Holographic/Quantum Neural Technology (HNeT) Quantum mechanical concepts of enfolding and quantum parallelism may be applied not only to energy and matter, but describe how single neuron cells are capable of learning vast amounts of stimulus-response information in real time - enfolding this information onto the same storage structures (i.e. computer RAM). It is now realized and accepted that a single holographic/quantum neural neuron cell is
capable of learning stimulus-response patterns or "memories" orders of magnitude faster, and far more accurately than the traditional back-propagation or genetically based neural networks.

Perus published his research at Neural Network World about his idea about Neural Networks as a basis for Quantum Associate Networks (Perus, 2000) and Quantum system can realize content_addressable associative memory (Perus and Dey, 2000).

Hu [2001] developed a research about Quantum computation via Neural Networks applied to Image processing and pattern recognition. He proved that the error in measurement produced by quantum principle is half the error produced by a classical approach.

A neural-net-like model, which is realizable using quantum holography, is proposed for quantum associative me-mory and pattern recognition. This Hopfield-based math-ematical model/algorithm, translated to quantum formalism, has been successfully tested in computer simulations of concrete pattern-recognition applications [Peru and Bischof, 2002].

The paper "Quantum-implemented selective reconstruction of high-resolution images" proposes quantum image reconstruction. Input-triggered selection of an image among many stored ones, and its reconstruction if the input is occluded or noisy, has been simulated by a computer program implementable in a real quantum-physical system [Peru,2004].

Quantum teleportation is surprising the scientific community since it can transmit an unknown quantum state following the no cloning principle of quantum mechanics [Ben93, Bou97]. During teleportation, some limited information is transmitted in the classical channel and the quantum state is transmitted in the quantum channel. Even though the original quantum state can be reconstructed (with the price of destroying the original state), nothing is learned during the transmission about the original state [Men98]. Considering the powerful learning ability of Artificial Neural Networks (ANN) [Hec90, Hye94], Entangled Neural Networks (ENNs) are proposed in this sense to learn information from inside and outside the system and to interfere and manipulate the huge amount of knowledge using quantum parallelism. There is some interesting literature about the use of ANN with quantum computing [Chr95, Kak95, Men95, Zar95, Beh96, Pru96, Val96, Men98, Van98, Wei98], and especially, a more recent review from Hirsh et al. [Hir99, Hog99, Kak99, Ven99].

Entangled Neural Networks (ENNs) [Wei99, to get full paper in pdf] are proposed on the basis of quantum teleportation and its extension to intelligent sense. Neuron A (Alice), neuron B (Bob), an EPR source [Ben93] and some connections (classical and quantum channels) form a basic ENN (or unit). The operation of every ENN looks like quantum teleportation but the measurement of neuron A is orientated with intelligence. Connections in certain manners among these units construct ENN. With this arrangement, the simulation of whole ENN tries to follow a general hypothesis in which the conscious decision is stimulated by the influences of a lot of unconscious factors [Pen89, Teg99]. In ENNs, there is no repeated learning sequence as is the case with classical ANN, therefore, the decoherence problem may be reduced. We briefly describe the basic concepts of quantum teleportation and the development of ENNs. Through an example of temperature adjusting, we show the application of Hebb´s learning law and the decision sequence using ENNs.

Fourth International Conference on COMPUTATIONAL INTELLIGENCE AND NEUROSCIENCE: Special Sessions on
Quantum Computation and Neuro-Quantum Information Processing (Dan Ventura2000).

Since this page has published, a lot of researchers sent e-mails to me for recommending their researches as J. Wallace, Z.Z. Hu, and M. Perus. The author thanks deeply for their support.

The research is supported in part by grant # 521442/97-4 from CNPq, Brazil.

Some information in this paper do not get the original author´s personal agreement. I got their main points just as the common manner to get the the reference from science community. If there is any problem, please tell me.

8. References

[Bar96] Barenco, Adriano, "Quantum physics and computers", Contemporary Physics, vol.37, no. 5, pp.375-89, 1996.
[Beh96] Behrman, E. C., J. Niemel, J. E. Steck and S. R. Skinner, "A Quantum Dot Neural Network", IEEE Transactions on
Neural Networks, submitted, 1996.
[Ben93] Bennett, C. H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wooters, W., "Teleporting an Unknown Quantum
State via Dual Classical and Einstein-Podolsky-Rosen Channels", Physical Review Letters, Vol. 70 (1993) pp.1895-1899.

[Bie99] Bieberich, E. "Non-local quantum evolution of entangled ensemble states in neural nets and its significance for brain function and a theory of consciousness", 1999, quant-ph/9906011.
[Bou97] Bouwmeester, D., J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, “Experimental Quantum Teleportation” , Nature vol.390, pp.575, 1997.
[Chr95] Chrisley, R., "Quantum learning", in Pylkkänen, P. and Pylkkö, P., editors, New directions in cognitive science: Proceedings of the international symposium, Saariselka, 4-9 August 1995, Lapland, Finland, pages 77-89, Helsinki. Finnish Association of Artificial Intelligence.
[Deu89] Deutsch, D. "Quantum computational networks", in Proceedings of the Royal Society of London A, vol. 425, pp.
73-90, 1989.
[Eke98] Ekert, A. and D. Deutsch: Quantum Computation, the special "quantum information", issue of Physics World, March
1998.
[Fey82] Feynman, R., "Simulating physics with computers", International Journal of Theoretical Physics, 21:467-488, 1982.
[Gros98] Grossberg, S., "Birth Of A Learning Law", INNS/ENNS/JNNS Newsletter, Neural Networks, Appearing with
Volume 11, No. 1, 1998.
[Grov96] Grover, L. K., "A fast quantum mechanical algorithm for database search", Proceedings of the 28th Annual ACM
Symposium on the Theory of Computing, ACM, New York, pp. 212-19, 1996.

[Lag97] Lagaris, I. E., A. Likas, D. I. Fotiadis "Artificial Neural Network Methods in Quantum Mechanics", Comput.Phys. Commun. 104 (1997) 1-14.
[Lug98] Luger, G. F. and W. A. Stubblefield. "Artificial Inteligente", Addison-Wesley Longman, Inc., Reading, MA, 1998.
[Hay94] Haykin, S., "Neural networks -- a comprehensive foundation", Macmillan College Publishing Company, Inc. Englewood Cliffs, NJ07632, USA, 1994.
[Hec90] Hecht-Nielsen, R., "Neurocomputing", Addison-Wesley Publishing Company, Reading, 1990.
[Hir99] Hirsh, H., “A Quantum leap for AI”, IEEE Intelligent System, pp. 9, July/August, 1999.
[Hog99] Hogg, T., “Quantum Search Heuristics”, IEEE Intelligent System, pp. 12-14, July/August, 1999.

[Hu2001] Hu, Z.Z., Quantum computation via Neural Networks applied to Image processing and pattern recognition. PhD thesis at the University of Western Sydney, Australia.
[Kak95] Kak, S. “Quantum Neural Computing”, Advances in Imaging and Electron Physics, vol. 94, pp. 259-313, 1995.
[Kak99] Kak, S. “Quantum Computing and AI”, IEEE Intelligent System, pp. 9-11, July/August, 1999.
[Men95] Menneer, T. and A. Narayanan, "Quantum-inspired Neural Networks", technical report R329, Department of Computer Science, University of Exeter, Exeter, United Kingdom, 1995.
[Men98] Menneer, T. “Quantum Artificial Neural Networks”, Ph. D. thesis of The University of Exeter, UK, May, 1998.
[Per96] Perus, M., "Neuro-Quantum Parallelism in Brain-Mind and Computers", Informatica, vol. 20, pp. 173-83, 1996.

[Perus97] M. Perus, Mind: neural computing plus quantum consciousness, in Mind Versus Computer, Edited by M. Gams, M. Paprzychi and X. Wu, by IOS press, pp. 156-170, 1997

[Perus98] Common mathematical foundations of neural and quantum informatics, in Zeitschrift Fur Angewandte Mathematik und Mechanik, Vol. 78 (1998), Suppl. 1, p9. 23-26. 

[Perus2000] Neural Networks as a basis for Quantum Associate Networks, in Neural Network World, vol. 10, n. 6, pp.1001-1013, 2000.

[Perus and Dey, 2000] Perus and Dey, Quantum system can realize content_addressable associative memory, in Applied Mathematics Letters, vol. 13, no. 8, pp.31-36, Pergamon, 2000.

[Perus and Bischof, 2002].Quantum-wave pattern recognition: from simulations towards implementation, Comput. Intellig. and Natural Comp. 2002 conf. in North Carolina. 

[Pes99] Pessa, E, G. Vitiello: "Quantum dissipation and neural net dynamics", 1999, quant-ph/9912070.
[Pur98] Purushothaman, G. and N.B. Karayiannis, Quantum neural networks (qnns) - inherently fuzzy feedforward neural networks, IEEE Transactions on Neural Networks, vol. 8, no. 3, pp.679-693, 1997.

[Per04] Quantum-implemented selective reconstruction of high-resolution images, Quantum Physics, abstract
quant-ph/0401016, http://arXiv.org/abs/quant-ph/0401016.
[Sho94] Shor, P. W., "Algorithms for quantum computation: Discrete log and factoring". In Goldwasser, S., editor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, pp. 124-134. IEEE Computer Society Press, 1994.

[Ven98] Ventura, D. and T. Martinez, "Quantum Associative Memory", preprint submitted to IEEE Transactions on Neural
Networks, June 16, 1998.
[Ven99] Ventura, D. , "Quantum Computational intelligence: Answers and Questions", IEEE Intelligent System, pp. 14-16, July/August, 1999.
[Vla97] Vlasov, A. Y. “Quantum Computations and Images Recognition”, e-print: http://xxx. lanl.gov/quant-ph/ 9703010,
1997.
[Wal99] Wallace, J. “Quantum Computer Simulators - A Review ”, School of Engineering and Computer Science, University of Exeter, UK, http://www.dcs.ex.ac.uk/~jwallace/simrevab.htm, 1999.
[Wal2000] Wallace, R. “Quantum linguisrics: information theory and quantum neural networks”, The New York State Psychiatic Institute, submitted for publication, Feb. 2000.
[Wei98] Weigang, L. “A study of parallel Self-Organizing Map”, e-print: http://xxx. lanl.gov/quant-ph/9808025, 1998.

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Initial publication : 1999; Last updated: 06-06-2003.