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{{short description|Quantum Mechanics in Neural Networks}}
[[File:Neural Network - basic scheme with legends.png|thumb|
'''Quantum neural networks''' are [[Neural network (machine learning)|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]] and Ron Chrisley,<ref>{{cite journal |first=S. |last=Kak |title=On quantum neural computing |journal=Advances in Imaging and Electron Physics |volume=94 |pages=259–313 |year=1995 |doi=10.1016/S1076-5670(08)70147-2 |isbn=9780120147366 }}</ref><ref>{{cite book |first=R. |last=Chrisley |chapter=Quantum Learning |title=New directions in cognitive science: Proceedings of the international symposium, Saariselka, 4–9 August 1995, Lapland, Finland |editor-first=P. |editor-last=Pylkkänen |editor2-first=P. |editor2-last=Pylkkö |publisher=Finnish Association of Artificial Intelligence |___location=Helsinki |pages=77–89 |year=1995 |isbn=951-22-2645-6 }}</ref> engaging with the theory of [[quantum mind]], which posits that quantum effects play a role in cognitive function. However, typical research in quantum neural networks involves combining classical [[artificial neural network]] models (which are widely used in machine learning for the important task of pattern recognition) with the advantages of [[quantum information]] in order to develop more efficient algorithms.<ref>{{cite journal|last1=da Silva|first1=Adenilton J.|last2=Ludermir|first2=Teresa B.|last3=de Oliveira|first3=Wilson R.|year=2016|title=Quantum perceptron over a field and neural network architecture selection in a quantum computer|journal=Neural Networks|volume=76|pages=55–64|arxiv=1602.00709|bibcode=2016arXiv160200709D|doi=10.1016/j.neunet.2016.01.002|pmid=26878722|s2cid=15381014}}</ref><ref>{{cite journal|last1=Panella|first1=Massimo|last2=Martinelli|first2=Giuseppe|year=2011|title=Neural networks with quantum architecture and quantum learning|journal=[[International Journal of Circuit Theory and Applications]]|volume=39|pages=61–77|doi=10.1002/cta.619|s2cid=3791858 }}</ref><ref>{{cite journal |first1=M. |last1=Schuld |first2=I. |last2=Sinayskiy |first3=F. |last3=Petruccione |arxiv=1408.7005 |title=The quest for a Quantum Neural Network |journal=Quantum Information Processing |volume=13 |issue=11 |pages=2567–2586 |year=2014 |doi=10.1007/s11128-014-0809-8 |bibcode=2014QuIP...13.2567S |s2cid=37238534 }}</ref> One important motivation for these investigations is the difficulty to train classical neural networks, especially in [[Big data|big data applications]]. The hope is that features of [[quantum computing]] such as [[quantum parallelism]] or the effects of [[quantum interference|interference]] and [[Quantum entanglement|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.
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=== 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
=== 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 operator|unitary]] [[quantum logic gate|gates]], or classically, via [[measurement in quantum mechanics|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 [[Integrated quantum photonics|photonically]] implemented neurons<ref name="WanDKGK16">{{cite journal|last1=Wan|first1=Kwok-Ho|last2=Dahlsten|first2=Oscar|last3=Kristjansson|first3=Hler|last4=Gardner|first4=Robert|last5=Kim|first5=Myungshik|year=2017|title=Quantum generalisation of feedforward neural networks|journal=npj Quantum Information|volume=3|issue=1 |pages=36|arxiv=1612.01045|bibcode=2017npjQI...3...36W|doi=10.1038/s41534-017-0032-4|s2cid=51685660}}</ref><ref>{{cite journal |first1=A. |last1=Narayanan |first2=T. |last2=Menneer |title=Quantum artificial neural network architectures and components |journal=Information Sciences |volume=128 |issue= 3–4|pages=231–255 |year=2000 |doi=10.1016/S0020-0255(00)00055-4 |s2cid=10901562 }}</ref> and [[quantum reservoir processor]] (quantum version of [[reservoir computing]]).<ref>{{cite journal |last1=Ghosh |first1=S. |last2=Opala |first2=A. |last3=Matuszewski |first3=M. |last4=Paterek |first4=P. |last5=Liew |first5=T. C. H. |doi=10.1038/s41534-019-0149-8 |title=Quantum reservoir processing |journal=npj Quantum Information |volume=5 |pages=35 |year=2019 |issue=1 |arxiv=1811.10335 |bibcode=2019npjQI...5...35G |s2cid=119197635 }}</ref> 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.<ref>{{cite arXiv |first1=H. |last1=Neven |display-authors=1 |first2=Vasil S. |last2=Denchev |first3=Geordie |last3=Rose |first4=William G. |last4=Macready |eprint=0811.0416 |title=Training a Binary Classifier with the Quantum Adiabatic Algorithm |year=2008 |class=quant-ph }}</ref>
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.<ref>{{cite journal |first1=J. |last1=Bang |display-authors=1 |first2=Junghee |last2=Ryu |first3=Seokwon |last3=Yoo |first4=Marcin |last4=Pawłowski |first5=Jinhyoung |last5=Lee |doi=10.1088/1367-2630/16/7/073017 |title=A strategy for quantum algorithm design assisted by machine learning |journal=New Journal of Physics |volume=16 |issue= 7|pages=073017 |year=2014 |arxiv=1301.1132 |bibcode=2014NJPh...16g3017B |s2cid=55377982 }}</ref><ref>{{cite journal |first1=E. C. |last1=Behrman |first2=J. E. |last2=Steck |first3=P. |last3=Kumar |first4=K. A. |last4=Walsh |arxiv=0808.1558 |title=Quantum Algorithm design using dynamic learning |journal=Quantum Information and Computation |volume=8 |issue=1–2 |pages=12–29 |year=2008 |doi=10.26421/QIC8.1-2-2 |s2cid=18587557 }}</ref> The quantum network thus ‘learns’ an algorithm.
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The first quantum associative memory algorithm was introduced by Dan Ventura and Tony Martinez in 1999.<ref>{{cite book |first1=D. |last1=Ventura |first2=T. |last2=Martinez |title=Artificial Neural Nets and Genetic Algorithms |chapter=A Quantum Associative Memory Based on Grover's Algorithm |chapter-url=https://pdfs.semanticscholar.org/d46f/e04b57b75a7f9c57f25d03d1c56b480ab755.pdf |archive-url=https://web.archive.org/web/20170911115617/https://pdfs.semanticscholar.org/d46f/e04b57b75a7f9c57f25d03d1c56b480ab755.pdf |url-status=dead |archive-date=2017-09-11 |pages=22–27 |year=1999 |doi=10.1007/978-3-7091-6384-9_5 |isbn=978-3-211-83364-3 |s2cid=3258510 }}</ref> The authors do not attempt to translate the structure of artificial neural network models into quantum theory, but propose an algorithm for a [[quantum circuit|circuit-based quantum computer]] that simulates [[associative memory (psychology)|associative memory]]. The memory states (in [[Hopfield neural network]]s saved in the weights of the neural connections) are written into a superposition, and a [[Grover search algorithm|Grover-like quantum search algorithm]] retrieves the memory state closest to a given input. As such, this is not a fully content-addressable memory, since only incomplete patterns can be retrieved.
The first truly content-addressable quantum memory, which can retrieve patterns also from corrupted inputs, was proposed by Carlo A. Trugenberger.<ref>{{Cite journal |last=Trugenberger |first=C. A. |date=2001-07-18 |title=Probabilistic Quantum Memories |url=http://dx.doi.org/10.1103/physrevlett.87.067901 |journal=Physical Review Letters |volume=87 |issue=6 |
Trugenberger,<ref name=":2" /> however, has shown that his
=== Classical neural networks inspired by quantum theory ===
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== Training ==
Quantum Neural Networks can be theoretically trained similarly to training classical/artificial neural networks. A key difference lies in communication between the layers of a neural networks. For classical neural networks, at the end of a given operation, the current [[perceptron]] copies its output to the next layer of perceptron(s) in the network. However, in a quantum neural network, where each perceptron is a qubit, this would violate the [[no-cloning theorem]].<ref name=":0" /><ref>{{Cite book|last1=Nielsen|first1=Michael A
Using this quantum feed-forward network, deep neural networks can be executed and trained efficiently. A deep neural network is essentially a network with many hidden-layers, as seen in the sample model neural network above. Since the Quantum neural network being discussed uses fan-out Unitary operators, and each operator only acts on its respective input, only two layers are used at any given time.<ref name=":0" /> In other words, no Unitary operator is acting on the entire network at any given time, meaning the number of qubits required for a given step depends on the number of inputs in a given layer. Since Quantum Computers are notorious for their ability to run multiple iterations in a short period of time, the efficiency of a quantum neural network is solely dependent on the number of qubits in any given layer, and not on the depth of the network.<ref name=":1" />
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This situation is known as Barren Plateaus, because most of the initial parameters are trapped on a "plateau" of almost zero gradient, which approximates random wandering<ref name=":3" /> rather than gradient descent. This makes the model untrainable.
In fact, not only QNN, but almost all deeper VQA algorithms have this problem. In the present [[Noisy intermediate-scale quantum era|NISQ era]], this is one of the problems that have to be solved if more applications are to be made of the various VQA algorithms, including QNN.
==See also==
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