# Tensor network

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{{Short description|Mathematical wave functions}}
alt=Two tensor networks|thumb|Two different tensor network representations of a single 7-indexed tensor (both networks can be contracted to it with 7 free indices remaining). The bottom one can be derived from the top one by performing contraction on the three 3-indexed tensors (in yellow) and merging them together.
'''Tensor networks''' or '''tensor network states''' are a class of variational [wave function](/source/wave_function)s used in the study of many-body quantum systems<ref name="Complex">{{Cite journal|last=Orús|first=Román|date=5 August 2019|title=Tensor networks for complex quantum systems|url=https://www.nature.com/articles/s42254-019-0086-7|journal=[Nature Reviews Physics](/source/Nature_Reviews_Physics)|language=en|volume=1|issue=9|pages=538–550|doi=10.1038/s42254-019-0086-7|issn=2522-5820|via=|arxiv=1812.04011|bibcode=2019NatRP...1..538O|s2cid=118989751}}</ref> and fluids.<ref name="Gourianov2022">{{Cite journal |last1=Gourianov |first1=Nikita |last2=Lubasch |first2=Michael |last3=Dolgov |first3=Sergey |last4=van den Berg |first4=Quincy Y. |last5=Babaee |first5=Hessam |last6=Givi |first6=Peyman |last7=Kiffner |first7=Martin |last8=Jaksch |first8=Dieter |title=A quantum-inspired approach to exploit turbulence structures |journal=Nature Computational Science |date=2022-01-01 |volume=2 |issue=1 |pages=30–37 |doi=10.1038/s43588-021-00181-1 |pmid=38177703 |issn=2662-8457|arxiv=2106.05782 }}</ref><ref>{{cite journal | last1 = Gourianov | first1 = Nikita | last2 = Givi | first2 = Peyman | last3 = Jaksch | first3 = Dieter | last4 = Pope | first4 = Stephen B. | title = Tensor networks enable the calculation of turbulence probability distributions | journal = Science Advances | volume = 11 | issue = 5 | article-number = eads5990 | year = 2025 | doi = 10.1126/sciadv.ads5990 | pmid = 39879287 | url = https://www.science.org/doi/abs/10.1126/sciadv.ads5990 | arxiv = 2407.09169 | bibcode = 2025SciA...11S5990G }}</ref> Tensor networks extend one-dimensional [matrix product state](/source/matrix_product_state)s to higher dimensions while preserving some of their useful mathematical properties.<ref name= "Practical">{{Cite journal|last=Orús|first=Román|date=2014-10-01|title=A practical introduction to tensor networks: Matrix product states and projected entangled pair states|url=http://www.sciencedirect.com/science/article/pii/S0003491614001596|journal=Annals of Physics|language=en|volume=349|pages=117–158|arxiv=1306.2164|doi=10.1016/j.aop.2014.06.013|bibcode=2014AnPhy.349..117O|s2cid=118349602|issn=0003-4916}}</ref>

The [wave function](/source/wave_function) is encoded as a [tensor contraction](/source/tensor_contraction) of a [network](/source/Graph_(discrete_mathematics)) of individual [tensor](/source/tensor)s.<ref>{{cite arXiv|last1=Biamonte|first1=Jacob|last2=Bergholm|first2=Ville|date=2017-07-31|title=Tensor Networks in a Nutshell|class=quant-ph|eprint=1708.00006}}</ref>  The structure of the individual tensors can impose global symmetries on the wave function (such as [antisymmetry under exchange](/source/Antisymmetrizer) of [fermion](/source/fermion)s) or restrict the wave function to specific [quantum number](/source/quantum_number)s, like total [charge](/source/Electric_charge), [angular momentum](/source/angular_momentum), or [spin](/source/Spin_(physics)).  It is also possible to derive strict bounds on quantities like [entanglement](/source/Quantum_entanglement) and [correlation length](/source/Correlation_function) using the mathematical structure of the tensor network.<ref>{{Cite journal|last1=Verstraete|first1=F.|last2=Wolf|first2=M. M.|last3=Perez-Garcia|first3=D.|last4=Cirac|first4=J. I.|date=2006-06-06|title=Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States|url=https://link.aps.org/doi/10.1103/PhysRevLett.96.220601|journal=Physical Review Letters|volume=96|issue=22|article-number=220601|doi=10.1103/PhysRevLett.96.220601|pmid=16803296|arxiv=quant-ph/0601075|bibcode=2006PhRvL..96v0601V|hdl=1854/LU-8590963|s2cid=119396305 |hdl-access=free}}</ref>  This has made tensor networks useful in theoretical studies of [quantum information](/source/quantum_information) in [many-body systems](/source/Many-body_problem).  They have also proved useful in [variational studies](/source/Variational_method_(quantum_mechanics)) of [ground state](/source/ground_state)s, [excited state](/source/excited_state)s, and [dynamics](/source/Time_evolution) of [strongly correlated many-body systems](/source/Strongly_correlated_material).<ref>{{Cite book|last=Montangero, Simone|title=Introduction to tensor network methods: numerical simulations of low-dimensional many-body quantum systems|date=28 November 2018|isbn=978-3-030-01409-4|location=Cham, Switzerland|oclc=1076573498}}</ref>

== Diagrammatic notation ==
In general, a [tensor network diagram (Penrose diagram)](/source/Penrose_graphical_notation) can be viewed as a [graph](/source/Graph_theory) where nodes (or vertices) represent individual tensors, while edges represent summation over an index. Free indices are depicted as edges (or ''legs'') attached to a single vertex only.<ref>{{Cite web |title=The Tensor Network |url=https://www.tensornetwork.org/ |access-date=2022-07-30 |website=Tensor Network |language=en}}</ref> Sometimes, there is also additional meaning to a node's shape. For instance, one can use trapezoids for unitary matrices or tensors with similar behaviour. This way, flipped trapezoids would be interpreted as complex conjugates to them.

== History ==
Foundational research on tensor networks began in 1971 with a paper by [Roger Penrose](/source/Roger_Penrose).<ref>Roger Penrose, "Applications of negative dimensional tensors," in ''Combinatorial Mathematics and its Applications'', Academic Press (1971). See Vladimir Turaev, ''Quantum invariants of knots and 3-manifolds'' (1994), De Gruyter, p. 71 for a brief commentary.</ref> In "Applications of negative dimensional tensors" Penrose developed [tensor diagram notation](/source/tensor_diagram_notation), describing how the diagrammatic language of tensor networks could be used in applications in physics.<ref name="Lectures">{{cite arXiv|last1=Biamonte|first1=Jacob|date=2020-04-01|title=Lectures on Quantum Tensor Networks |class=quant-ph|eprint=1912.10049}}</ref>

In 1992, [Steven R. White](/source/Steven_R._White) developed the [density matrix renormalization group](/source/density_matrix_renormalization_group) (DMRG) for quantum lattice systems.<ref name="DMRG">{{cite journal |last1=White|first1=Steven |title=Density matrix formulation for quantum renormalization groups |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.69.2863 |date=9 Nov 1992 |journal=Physical Review Letters |volume=69 |issue=19 |pages=2863–2866 |doi=10.1103/PhysRevLett.69.2863 |pmid=10046608 |bibcode=1992PhRvL..69.2863W |access-date=2024-10-24|url-access=subscription }}</ref><ref name= "Practical"></ref> The DMRG was the first successful tensor network and associated algorithm.<ref>{{cite web |title=Tensor Networks Group |url=https://www.simonsfoundation.org/mathematics-physical-sciences/many-electron-problem/tensor-networks/about/ |access-date=2024-10-24}}</ref>

In 2002, [Guifré Vidal](/source/Guifr%C3%A9_Vidal) and Reinhard Werner attempted to quantify entanglement, laying the groundwork for quantum resource theories.<ref>{{cite journal |last=Thomas |first=Jessica |date=2 Mar 2020 |title=50 Years of Physical Review A: The Legacy of Three Classics |journal=Physics |volume=13 |issue=<!---->|page=24 |doi=<!----> |bibcode=2020PhyOJ..13...24. |url=https://physics.aps.org/articles/v13/24 |access-date=2024-10-24}}</ref><ref>{{cite journal |last1=Vidal|first1=Guifre|last2=Werner|first2=Reinhard |title=Computable measure of entanglement |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.65.032314 |date=9 Nov 1992 |journal=Physical Review A |volume=65 |issue=3 |article-number=032314 |doi=10.1103/PhysRevA.65.032314 |access-date=2024-10-24|arxiv=quant-ph/0102117 }}</ref> This was also the first description of the use of tensor networks as mathematical tools for describing quantum systems.<ref name="Lectures"></ref>

In 2004, [Frank Verstraete](/source/Frank_Verstraete) and [Ignacio Cirac](/source/Ignacio_Cirac) developed the theory of matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems.<ref>{{cite journal |last1=Verstraete |first1=Frank |last2=Cirac |first2=Ignacio |title=Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems |url=https://www.tandfonline.com/doi/abs/10.1080/14789940801912366 |date=9 May 2007 |journal=Advances in Physics |volume=57 |issue=2 |page=143-224 |doi=10.1080/14789940801912366 |arxiv=0907.2796 |hdl=1854/LU-8589270 |access-date=2024-10-24}}</ref><ref name="Practical"></ref>

In 2006, Vidal developed the multi-scale entanglement renormalization ansatz (MERA).<ref>{{cite journal |last1=Vidal|first1=Guifre|last2=Werner|first2=Reinhard |title=Class of Quantum Many-Body States That Can Be Efficiently Simulated |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.110501 |date=12 Sep 2008 |journal=Physical Review Letters |volume=101 |issue=11 |article-number=110501 |doi=10.1103/PhysRevLett.101.110501 |pmid=18851269 |arxiv=quant-ph/0610099 |bibcode=2008PhRvL.101k0501V |access-date=2024-10-24}}</ref> In 2007 he developed entanglement renormalization for quantum lattice systems.<ref>{{cite arXiv|last1=Vidal|first1=Guifre |date=2009-12-09|title=Entanglement Renormalization: an introduction |class=quant-ph|eprint=0912.1651}}</ref>

In 2010, Ulrich Schollwock developed the density-matrix renormalization group for the simulation of one-dimensional strongly correlated quantum lattice systems.<ref>{{cite journal |last1=Schollwock |first1=Ulrich |title=The density-matrix renormalization group in the age of matrix product states |url=https://www.sciencedirect.com/science/article/abs/pii/S0003491610001752 |date=20 Aug 2010 |journal=Annals of Physics |volume=326 |issue=1 |page=96-192 |doi=10.1016/j.aop.2010.09.012 |arxiv=1008.3477 |access-date=2024-10-24}}</ref>

In 2014, [Román Orús](/source/Rom%C3%A1n_Or%C3%BAs) introduced tensor networks for complex quantum systems and machine learning, as well as tensor network theories of symmetries, fermions, entanglement and holography.<ref name="Complex"></ref><ref>{{cite journal |last1=Orús |first1=Román |title=Advances on tensor network theory: symmetries, fermions, entanglement, and holography |url=https://link.springer.com/article/10.1140/epjb/e2014-50502-9 |date=26 Nov 2014 |journal=The European Physical Journal B |volume=87 |issue=280 |article-number=280 |doi=10.1140/epjb/e2014-50502-9 |arxiv=1407.6552 |bibcode=2014EPJB...87..280O |access-date=2024-10-24}}</ref>

== Connection to machine learning ==
Tensor networks have been adapted for [supervised learning](/source/supervised_learning),<ref>{{cite journal|last1=Stoudenmire|first1=E. Miles|last2=Schwab|first2=David J.|date=2017-05-18|title=Supervised Learning with Quantum-Inspired Tensor Networks|journal=Advances in Neural Information Processing Systems|volume=29|page=4799|arxiv=1605.05775}}</ref> taking advantage of similar mathematical structure in [variational studies](/source/Variational_method_(quantum_mechanics)) in quantum mechanics and large-scale [machine learning](/source/machine_learning).  This crossover has spurred collaboration between researchers in [artificial intelligence](/source/artificial_intelligence) and [quantum information science](/source/quantum_information_science).  In June 2019, [Google](/source/Google), the [Perimeter Institute for Theoretical Physics](/source/Perimeter_Institute_for_Theoretical_Physics), and [X (company)](/source/X_Development), released TensorNetwork,<ref>{{Citation|title=google/TensorNetwork|date=2021-01-30|url=https://github.com/google/TensorNetwork|access-date=2021-02-02}}</ref> an open-source library for efficient tensor calculations.<ref>{{Cite web|title=Introducing TensorNetwork, an Open Source Library for Efficient Tensor Calculations|url=http://ai.googleblog.com/2019/06/introducing-tensornetwork-open-source.html|access-date=2021-02-02|website=Google AI Blog|date=4 June 2019 |language=en}}</ref>

The main interest in tensor networks and their study from the perspective of machine learning is to reduce the number of trainable parameters (in a layer) by approximating a high-order tensor with a network of lower-order ones. Using the so-called tensor train technique (TT),<ref>{{Cite journal |last=Oseledets |first=I. V. |date=2011-01-01 |title=Tensor-Train Decomposition |url=https://epubs.siam.org/doi/10.1137/090752286 |journal=SIAM Journal on Scientific Computing |volume=33 |issue=5 |pages=2295–2317 |doi=10.1137/090752286 |bibcode=2011SJSC...33.2295O |s2cid=207059098 |issn=1064-8275|url-access=subscription }}</ref> one can reduce an N-order tensor (containing exponentially many trainable parameters) to a chain of N tensors of order 2 or 3, which gives us a polynomial number of parameters.
thumb|Tensor train technique

== See also ==
* [Tensor](/source/Tensor)
* [Tensor diagrams](/source/Penrose_graphical_notation)
* [Tensor contraction](/source/Tensor_contraction)
* [Tensor Processing Unit (TPU)](/source/Tensor_Processing_Unit)
* [Tensor rank decomposition](/source/Tensor_rank_decomposition)
* [Einstein Notation](/source/Einstein_notation)
* [Spin network](/source/Spin_network)

==References==
{{Reflist}}

==External links==
*[https://tensornetwork.org tensornetwork.org - a resource for tensor network algorithms, theory, and software]
*[https://tensors.net tensors.net - tensor network tutorials, sample implementations and other resources]
*[https://link.springer.com/book/10.1007/978-3-030-34489-4 Tensor Network Contractions: Methods and Applications to Quantum Many-Body Systems]
*[https://tensor4all.org/ tensor4all - website]

Category:Applied mathematics
Category:Concepts in physics
Category:Quantum states
Category:Applications of artificial intelligence
Category:Lattice field theory

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Adapted from the Wikipedia article [Tensor network](https://en.wikipedia.org/wiki/Tensor_network) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Tensor_network?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.