Quantum neural network: Difference between revisions

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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 <ref>{{cite journal |first=M. |last=Perus |title=Neural Networks as a basis for quantum associative memory |journal=Neural Network World |volume=10 |issue=6 |pages=1001 |year=2000 |citeseerx=10.1.1.106.4583 }}</ref><ref>{{cite journal |first1=M. |last1=Zak |first2=C. P. |last2=Williams |title=Quantum Neural Nets |journal=International Journal of Theoretical Physics |volume=37 |issue=2 |pages=651–684 |year=1998 |doi=10.1023/A:1026656110699 |s2cid=55783801 }}</ref> to postulating non-linear quantum operators (a mathematical framework that is disputed).<ref>{{Cite journal | doi=10.1006/jcss.2001.1769| title=Quantum Neural Networks| journal=Journal of Computer and System Sciences| volume=63| issue=3| pages=355–383| year=2001| last1=Gupta| first1=Sanjay| last2=Zia| first2=R.K.P.| arxiv=quant-ph/0201144| s2cid=206569020}}</ref><ref>{{cite journalweb |first1=J. |last1=Faber |first2=G. A. |last2=Giraldi |title=Quantum Models for Artificial Neural Network |year=2002 |url=http://arquivoswebvirtual01.lncc.br/pdfs~giraldi/TechReport/QNN-Review.pdf }}{{Dead link|date=May 2024 |bot=InternetArchiveBot |fix-attempted=yes.gz }}</ref> A direct implementation of the activation function using the [[quantum circuit|circuit-based model of quantum computation]] has recently been proposed by Schuld, Sinayskiy and Petruccione based on the [[quantum phase estimation algorithm]].<ref>{{cite journal |first1=M. |last1=Schuld |first2=I. |last2=Sinayskiy |first3=F. |last3=Petruccione |title=Simulating a perceptron on a quantum computer |journal=Physics Letters A |arxiv=1412.3635 |year=2014 |volume=379 |issue=7 |pages=660–663 |doi=10.1016/j.physleta.2014.11.061 |s2cid=14288234 }}</ref>

=== 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|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 |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 journalarxiv |first1=H. |last1=Neven |display-authors=1 |first2=Vasil S. |last2=Denchev |first3=Geordie |last3=Rose |first4=William G. |last4=Macready |arxiv=0811.0416 |title=Training a Binary Classifier with the Quantum Adiabatic Algorithm |year=2008 }}</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.