Tensor network

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]

Two tensor networks
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.

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]

  1. ^ 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.
  2. ^ 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.
  3. ^ 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].
  4. ^ 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.
  5. ^ Biamonte, Jacob; Bergholm, Ville (2017-07-31). "Tensor Networks in a Nutshell". arXiv:1708.00006 [quant-ph].
  6. ^ 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.
  7. ^ 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)