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] Tensor networks extend one-dimensional matrix product states to higher dimensions while preserving some of their useful mathematical properties.[2]

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.[3] 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.[4] 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.[5]

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.[6] 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.

Connection to machine learning

Tensor networks have been adapted for supervised learning,[7] 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,[8] an open-source library for efficient tensor calculations.[9]

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),[10] 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.

Tensor train technique

See also

References
  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. 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.
  3. Biamonte, Jacob; Bergholm, Ville (2017-07-31). "Tensor Networks in a Nutshell". arXiv:1708.00006 [quant-ph].
  4. 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.
  5. 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)
  6. "The Tensor Network". Tensor Network. Retrieved 2022-07-30.
  7. 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.
  8. google/TensorNetwork, 2021-01-30, retrieved 2021-02-02
  9. "Introducing TensorNetwork, an Open Source Library for Efficient Tensor Calculations". Google AI Blog. 4 June 2019. Retrieved 2021-02-02.
  10. 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.
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