Tensor network
{{Short description|Mathematical wave functions}}
File:Tensor network contraction example.png
Tensor networks or tensor network states are a class of variational wave functions used in the study of many-body quantum systems{{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|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}} and fluids.{{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 |url=https://doi.org/10.1038/s43588-021-00181-1 |issn=2662-8457|arxiv=2106.05782 }}{{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 | page = 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 }} Tensor networks extend one-dimensional matrix product states to higher dimensions while preserving some of their useful mathematical properties.{{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}}
The wave function is encoded as a tensor contraction of a network of individual tensors.{{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}} The structure of the individual tensors can impose global symmetries on the wave function (such as antisymmetry under exchange of fermions) or restrict the wave function to specific quantum numbers, like total charge, angular momentum, or spin. It is also possible to derive strict bounds on quantities like entanglement and correlation length using the mathematical structure of the tensor network.{{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|pages=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}} This has made tensor networks useful in theoretical studies of quantum information in many-body systems. They have also proved useful in variational studies of ground states, excited states, and dynamics of strongly correlated many-body systems.{{Cite book|last=Montangero, Simone|url=https://www.worldcat.org/oclc/1076573498|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}}
Diagrammatic notation
In general, a tensor network diagram (Penrose diagram) can be viewed as a graph 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.{{Cite web |title=The Tensor Network |url=https://www.tensornetwork.org/ |access-date=2022-07-30 |website=Tensor Network |language=en}} 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.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. In “Applications of negative dimensional tensors” Penrose developed tensor diagram notation, describing how the diagrammatic language of tensor networks could be used in applications in physics.{{cite arXiv|last1=Biamonte|first1=Jacob|date=2020-04-01|title=Lectures on Quantum Tensor Networks |class=quant-ph|eprint=1912.10049}}
In 1992, Steven R. White developed the Density matrix renormalization group (DMRG) for quantum lattice systems.{{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 }} The DMRG was the first successful tensor network and associated algorithm.{{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}}
In 2002, Guifré Vidal and Reinhard Werner attempted to quantify entanglement, laying the groundwork for quantum resource theories.{{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=|pages=24 |doi= |bibcode=2020PhyOJ..13...24. |url=https://physics.aps.org/articles/v13/24 |access-date=2024-10-24}}{{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 |page=032314 |doi=10.1103/PhysRevA.65.032314 |access-date=2024-10-24|arxiv=quant-ph/0102117 }} This was also the first description of the use of tensor networks as mathematical tools for describing quantum systems.
In 2004, Frank Verstraete and Ignacio Cirac developed the theory of matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems.{{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}}
In 2006, Vidal developed the multi-scale entanglement renormalization ansatz (MERA).{{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 |page=110501 |doi=10.1103/PhysRevLett.101.110501 |pmid=18851269 |arxiv=quant-ph/0610099 |bibcode=2008PhRvL.101k0501V |access-date=2024-10-24}} In 2007 he developed entanglement renormalization for quantum lattice systems.{{cite arXiv|last1=Vidal|first1=Guifre |date=2009-12-09|title=Entanglement Renormalization: an introduction |class=quant-ph|eprint=0912.1651}}
In 2010, Ulrich Schollwock developed the density-matrix renormalization group for the simulation of one-dimensional strongly correlated quantum lattice systems.{{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}}
In 2014, Román Orús introduced tensor networks for complex quantum systems and machine learning, as well as tensor network theories of symmetries, fermions, entanglement and holography.{{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 |doi=10.1140/epjb/e2014-50502-9 |arxiv=1407.6552 |bibcode=2014EPJB...87..280O |access-date=2024-10-24}}
Connection to machine learning
Tensor networks have been adapted for supervised learning,{{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}} taking advantage of similar mathematical structure in variational studies in quantum mechanics and large-scale machine learning. This crossover has spurred collaboration between researchers in artificial intelligence and quantum information science. In June 2019, Google, the Perimeter Institute for Theoretical Physics, and X (company), released TensorNetwork,{{Citation|title=google/TensorNetwork|date=2021-01-30|url=https://github.com/google/TensorNetwork|access-date=2021-02-02}} an open-source library for efficient tensor calculations.{{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}}
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),{{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 }} 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.
See also
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]