Quantum neural network

Quantum neural networks (QNNs) are computational neural network models which are based on the principles of quantum mechanics. The first ideas on quantum neural computation were published independently in 1995 by Subhash Kak[1] and Ron Chrisley[2], engaging with the theory of quantum mind, which posits that quantum effects play a role in cognitive function. However, typical research in QNNs involves combining classical artificial neural network models (which are widely used in machine learning for the important task of pattern classification) with the advantages of quantum information in order to develop more efficient algorithms [3][4][5]. One important motivation for these investigations is the difficulty to train classical neural networks, especially in big data applications. The hope is that features of quantum computing such as quantum parallelism or the effects of interference and entanglement can be used as resources. Since the technological implementation of a quantum computer is still in a premature stage, such quantum neural network models are mostly theoretical proposals that await their full implementation in physical experiments.


Quantum neural network research is still in its infancy, and a conglomeration of proposals and ideas of varying scope and mathematical rigor have been put forward. Most of them are based on the idea of replacing classical binary or McCulloch-Pitts neurons with a qubit (which can be called a “quron”), resulting in neural units that can be in a superposition of the state ‘firing’ and ‘resting’.

Quantum perceptrons

A lot of proposals attempt to find a quantum equivalent for the perceptron unit from which neural nets are constructed. A problem is that nonlinear activation functions do not immediately correspond to the mathematical structure of quantum theory, since a quantum evolution is described by linear operations and leads to probabilistic observation. Ideas to imitate the perceptron activation function with a quantum mechanical formalism reach from special measurements [6][7] to postulating non-linear quantum operators (a mathematical framework that is disputed).[8][9] A direct implementation of the activation function using the circuit-based model of quantum computation has recently been proposed by Schuld, Sinayskiy and Petruccione based on the quantum phase estimation algorithm.[10]

Quantum networks

At a larger scale, researchers have attempted to generalize neural networks to the quantum setting. One way of constructing a quantum neuron is to first generalise classical neurons and then generalising them further to make unitary gates. Interactions between neurons can be controlled quantumly, with unitary gates, or classically, via measurement of the network states. This high-level theoretical technique can be applied broadly, by taking different types of networks and different implementations of quantum neurons, such as photonically implemented neurons[11][12] and quantum reservoir processor.[13] Most learning algorithms follow the classical model of training an artificial neural network to learn the input-output function of a given training set and use classical feedback loops to update parameters of the quantum system until they converge to an optimal configuration. Learning as a parameter optimisation problem has also been approached by adiabatic models of quantum computing.[14]

Quantum neural networks can be applied to algorithmic design: given qubits with tunable mutual interactions, one can attempt to learn interactions following the classical backpropagation rule from a training set of desired input-output relations, taken to be the desired output algorithm's behavior.[15][16] The quantum network thus ‘learns’ an algorithm.

Quantum associative memory

The quantum associative memory algorithm[17] was introduced by Dan Ventura and Tony Martinez in 1999. The authors do not attempt to translate the structure of artificial neural network models into quantum theory, but propose an algorithm for a circuit-based quantum computer that simulates associative memory. The memory states (in Hopfield neural networks saved in the weights of the neural connections) are written into a superposition, and a Grover-like quantum search algorithm retrieves the memory state closest to a given input. An advantage lies in the exponential storage capacity of memory states, however the question remains whether the model has significance regarding the initial purpose of Hopfield models as a demonstration of how simplified artificial neural networks can simulate features of the brain.

Classical neural networks inspired by quantum theory

A substantial amount of interest has been given to a “quantum-inspired” model that uses ideas from quantum theory to implement a neural network based on fuzzy logic.[18]

See also


  1. S. Kak, On quantum neural computing, Advances in Imaging and Electron Physics 94, 259 (1995)
  2. R. Chrisley, Quantum Learning, In New directions in cognitive science: Proceedings of the international symposium, Saariselka, 4–9 August 1995, Lapland, Finland, P. Pylkkänen and P. Pylkkö (editors). Finnish Association of Artificial Intelligence, Helsinki, 77-89 (1995)
  3. da Silva, Adenilton J.; Ludermir, Teresa B.; de Oliveira, Wilson R. (2016). "Quantum perceptron over a field and neural network architecture selection in a quantum computer". Neural Networks. 76: 55–64. arXiv:1602.00709. Bibcode:2016arXiv160200709D. doi:10.1016/j.neunet.2016.01.002. PMID 26878722.
  4. Panella, Massimo; Martinelli, Giuseppe (2011). "Neural networks with quantum architecture and quantum learning". International Journal of Circuit Theory and Applications. 39: 61–77. doi:10.1002/cta.619.
  5. M. Schuld, I. Sinayskiy, F. Petruccione: The quest for a Quantum Neural Network, Quantum Information Processing 13, 11, pp. 2567-2586 (2014)
  6. M. Perus: Neural Networks as a basis for quantum associative memory, Neural Network World 10 (6), 1001 (2000)
  7. M. Zak, C.P. Williams: Quantum Neural Nets, International Journal of Theoretical Physics 37(2), 651 (1998)
  8. Gupta, Sanjay; Zia, R.K.P. (2001). "Quantum Neural Networks". Journal of Computer and System Sciences. 63 (3): 355–383. arXiv:quant-ph/0201144. doi:10.1006/jcss.2001.1769.
  9. J. Faber, G.A. Giraldi: Quantum Models for Artificial Neural Network (2002), Electronically available: http://arquivosweb.%5B%5D lncc.br/pdfs/QNN-Review. pdf
  10. M. Schuld, I. Sinayskiy, F. Petruccione: Simulating a perceptron on a quantum computer ArXiv:1412.3635 (2014)
  11. Wan, Kwok-Ho; Dahlsten, Oscar; Kristjansson, Hler; Gardner, Robert; Kim, Myungshik (2017). "Quantum generalisation of feedforward neural networks". NPJ Quantum Information. 3: 36. arXiv:1612.01045. Bibcode:2017npjQI...3...36W. doi:10.1038/s41534-017-0032-4.
  12. A. Narayanan and T. Menneer: Quantum artificial neural network architectures and components, Information Sciences 128, 231-255 (2000)
  13. Ghosh, S.; Opala, A.; Matuszewski, M.; Paterek, P; Liew, T. C. H.: Quantum reservoir processing, Npj Quant. Info. 5, 35 (2019)
  14. H. Neven et al.: Training a Binary Classifier with the Quantum Adiabatic Algorithm, arXiv:0811.0416v1 (2008)
  15. J. Bang et al. : A strategy for quantum algorithm design assisted by machine learning, New Journal of Physics 16 073017 (2014)
  16. E.C. Behrman, J.E. Steck, P. Kumar, K.A. Walsh: Quantum Algorithm design using dynamic learning, Quantum Information and Computation, vol. 8, No. 1&2, pp. 12-29 (2008)
  17. D. Ventura, T. Martinez: A quantum associative memory based on Grover's algorithm, Proceedings of the International Conference on Artificial Neural Networks and Genetics Algorithms, pp. 22-27 (1999)
  18. G. Purushothaman, N. Karayiannis: Quantum Neural Networks (QNN’s): Inherently Fuzzy Feedforward Neural Networks, IEEE Transactions on Neural Networks, 8(3), 679 (1997)
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