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{{short description|Quantum Mechanics in Neural Networks}}
[[File:Neural Network - basic scheme with legends.png|thumb|
'''Quantum neural networks''' are
Most Quantum neural networks are developed as [[Feedforward neural network|feed-forward]] networks. Similar to their classical counterparts, this structure intakes input from one layer of qubits, and passes that input onto another layer of qubits. This layer of qubits evaluates this information and passes on the output to the next layer. Eventually the path leads to the final layer of qubits.<ref name=":0">{{Cite journal|last1=Beer|first1=Kerstin|last2=Bondarenko|first2=Dmytro|last3=Farrelly|first3=Terry|last4=Osborne|first4=Tobias J.|last5=Salzmann|first5=Robert|last6=Scheiermann|first6=Daniel|last7=Wolf|first7=Ramona|date=2020-02-10|title=Training deep quantum neural networks|url= |journal=Nature Communications|language=en|volume=11|issue=1|pages=808|doi=10.1038/s41467-020-14454-2|issn=2041-1723|pmc=7010779|pmid=32041956|arxiv=1902.10445|bibcode=2020NatCo..11..808B}}</ref><ref name="WanDKGK16" /> The layers do not have to be of the same width, meaning they don't have to have the same number of qubits as the layer before or after it. This structure is trained on which path to take similar to classical [[artificial neural network]]s. This is discussed in a lower section. Quantum neural networks refer to three different categories: Quantum computer with classical data, classical computer with quantum data, and quantum computer with quantum data.<ref name=":0" />
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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
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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Equation 2 <math>C ={1 \over N}\sum_{x}^N{\langle\phi^\text{out}|\rho^\text{out}|\phi^\text{out}\rangle}</math>
=== Barren plateaus ===
[[File:Barren_plateaus_of_VQA.webp|alt=The Barren Plateau problem becomes increasingly serious as the VQA expands|thumb|'''Barren plateaus of VQA'''<ref>{{Cite journal |last1=Wang |first1=Samson |last2=Fontana |first2=Enrico |last3=Cerezo |first3=M. |last4=Sharma |first4=Kunal |last5=Sone |first5=Akira |last6=Cincio |first6=Lukasz |last7=Coles |first7=Patrick J. |date=2021-11-29 |title=Noise-induced barren plateaus in variational quantum algorithms |journal=Nature Communications |language=en |volume=12 |issue=1 |page=6961 |arxiv=2007.14384 |bibcode=2021NatCo..12.6961W |doi=10.1038/s41467-021-27045-6 |issn=2041-1723 |pmc=8630047 |pmid=34845216}}</ref> Figure shows the Barren Plateau problem becomes increasingly serious as the VQA expands.]]
Gradient descent is widely used and successful in classical algorithms. However, although the simplified structure is very similar to neural networks such as CNNs, QNNs perform much worse.
Since the quantum space exponentially expands as the q-bit grows, the observations will concentrate around the mean value at an exponential rate, where also have exponentially small gradients.<ref name=":3">{{Cite journal |last1=McClean |first1=Jarrod R. |last2=Boixo |first2=Sergio |last3=Smelyanskiy |first3=Vadim N. |last4=Babbush |first4=Ryan |last5=Neven |first5=Hartmut |date=2018-11-16 |title=Barren plateaus in quantum neural network training landscapes |journal=Nature Communications |language=en |volume=9 |issue=1 |page=4812 |arxiv=1803.11173 |bibcode=2018NatCo...9.4812M |doi=10.1038/s41467-018-07090-4 |issn=2041-1723 |pmc=6240101 |pmid=30446662}}</ref>
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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[[Category:Artificial neural networks]]
[[Category:Quantum information science]]
[[Category:Quantum programming]]
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