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**Tensor networks** or **tensor network states** are a class of variational wave functions used in the study of many-body quantum systems^{[1]} and fluids.^{[2]}^{[3]} Tensor networks extend one-dimensional matrix product states to higher dimensions while preserving some of their useful mathematical properties.^{[4]}

The wave function is encoded as a tensor contraction of a network of individual tensors.^{[5]} 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.^{[6]} 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.^{[7]}

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.^{[8]} 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.

Foundational research on tensor networks began in 1971 with a paper by Roger Penrose.^{[9]} 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.^{[10]}

In 1992, Steven R. White developed the Density Matrix Renormalization Group (DMRG) for quantum lattice systems.^{[11]}^{[4]} The DMRG was the first successful tensor network and associated algorithm.^{[12]}

In 2002, Guifre Vidal and Reinhard Werner attempted to quantify entanglement, laying the groundwork for quantum resource theories.^{[13]}^{[14]} This was also the first description of the use of tensor networks as mathematical tools for describing quantum systems.^{[10]}

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.^{[15]}^{[4]}

In 2006, Vidal developed the multi-scale entanglement renormalization ansatz (MERA).^{[16]} In 2007 he developed entanglement renormalization for quantum lattice systems.^{[17]}

In 2010, Ulrich Schollwock developed the density-matrix renormalization group for the simulation of one-dimensional strongly correlated quantum lattice systems.^{[18]}

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.^{[1]}^{[19]}

Tensor networks have been adapted for supervised learning,^{[20]} 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,^{[21]} an open-source library for efficient tensor calculations.^{[22]}

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),^{[23]} 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.

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^{a}^{b}Orús, Román (5 August 2019). "Tensor networks for complex quantum systems".*Nature Reviews Physics*.**1**(9): 538–550. arXiv:1812.04011. Bibcode:2019NatRP...1..538O. doi:10.1038/s42254-019-0086-7. ISSN 2522-5820. S2CID 118989751. **^**Gourianov, Nikita; Lubasch, Michael; Dolgov, Sergey; van den Berg, Quincy Y.; Babaee, Hessam; Givi, Peyman; Kiffner, Martin; Jaksch, Dieter (2022-01-01). "A quantum-inspired approach to exploit turbulence structures".*Nature Computational Science*.**2**(1): 30–37. doi:10.1038/s43588-021-00181-1. ISSN 2662-8457. PMID 38177703.**^**Gourianov, Nikita; Givi, Peyman; Jaksch, Dieter; Pope, Stephen B. (2024). "Tensor networks enable the calculation of turbulence probability distributions". arXiv:2407.09169 [physics.flu-dyn].- ^
^{a}^{b}^{c}Orús, Román (2014-10-01). "A practical introduction to tensor networks: Matrix product states and projected entangled pair states".*Annals of Physics*.**349**: 117–158. arXiv:1306.2164. Bibcode:2014AnPhy.349..117O. doi:10.1016/j.aop.2014.06.013. ISSN 0003-4916. S2CID 118349602. **^**Biamonte, Jacob; Bergholm, Ville (2017-07-31). "Tensor Networks in a Nutshell". arXiv:1708.00006 [quant-ph].**^**Verstraete, F.; Wolf, M. M.; Perez-Garcia, D.; Cirac, J. I. (2006-06-06). "Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States".*Physical Review Letters*.**96**(22): 220601. arXiv:quant-ph/0601075. Bibcode:2006PhRvL..96v0601V. doi:10.1103/PhysRevLett.96.220601. hdl:1854/LU-8590963. PMID 16803296. S2CID 119396305.**^**Montangero, Simone (28 November 2018).*Introduction to tensor network methods : numerical simulations of low-dimensional many-body quantum systems*. Cham, Switzerland. ISBN 978-3-030-01409-4. OCLC 1076573498.`{{cite book}}`

: CS1 maint: location missing publisher (link)**^**"The Tensor Network".*Tensor Network*. Retrieved 2022-07-30.**^**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.- ^
^{a}^{b}Biamonte, Jacob (2020-04-01). "Lectures on Quantum Tensor Networks". arXiv:1912.10049 [quant-ph]. **^**White, Steven (9 Nov 1992). "Density matrix formulation for quantum renormalization groups".*Physical Review Letters*.**69**(19). doi:10.1103/PhysRevLett.69.2863. Retrieved 2024-10-24.**^**"Tensor Networks Group". Retrieved 2024-10-24.**^**Thomas, Jessica (2 Mar 2020). "50 Years of Physical Review A: The Legacy of Three Classics". Retrieved 2024-10-24.**^**Vidal, Guifre; Werner, Reinhard (9 Nov 1992). "Computable measure of entanglement".*Physical Review Letters A*.**65**(3). doi:10.1103/PhysRevA.65.032314. Retrieved 2024-10-24.**^**Verstraete, Frank; Cirac, Ignacio (9 May 2007). "Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems".*Advances in Physics*.**57**(2): 143-224. arXiv:0907.2796. doi:10.1080/14789940801912366. Retrieved 2024-10-24.**^**Vidal, Guifre; Werner, Reinhard (12 Sep 2008). "Class of Quantum Many-Body States That Can Be Efficiently Simulated".*Physical Review Letters*.**101**(11). arXiv:quant-ph/0610099. doi:10.1103/PhysRevLett.101.110501. Retrieved 2024-10-24.**^**Vidal, Guifre (2009-12-09). "Entanglement Renormalization: an introduction". arXiv:0912.1651 [quant-ph].**^**Schollwock, Ulrich (20 Aug 2010). "The density-matrix renormalization group in the age of matrix product states".*Annals of Physics*.**326**(1): 96-192. arXiv:1008.3477. doi:10.1016/j.aop.2010.09.012. Retrieved 2024-10-24.**^**Orús, Román (26 Nov 2014). "Advances on tensor network theory: symmetries, fermions, entanglement, and holography".*The European Physical Journal B*.**87**(280). arXiv:1407.6552. doi:10.48550/arXiv.1407.6552. Retrieved 2024-10-24.**^**Stoudenmire, E. Miles; Schwab, David J. (2017-05-18). "Supervised Learning with Quantum-Inspired Tensor Networks".*Advances in Neural Information Processing Systems*.**29**: 4799. arXiv:1605.05775.**^***google/TensorNetwork*, 2021-01-30, retrieved 2021-02-02**^**"Introducing TensorNetwork, an Open Source Library for Efficient Tensor Calculations".*Google AI Blog*. 4 June 2019. Retrieved 2021-02-02.**^**Oseledets, I. V. (2011-01-01). "Tensor-Train Decomposition".*SIAM Journal on Scientific Computing*.**33**(5): 2295–2317. Bibcode:2011SJSC...33.2295O. doi:10.1137/090752286. ISSN 1064-8275. S2CID 207059098.