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Tensor networks for complex quantum systems

Nature Reviews Physics volume 1, pages 538–550 (2019)Cite this article

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Abstract

Originally developed in the context of condensed-matter physics and based on renormalization group ideas, tensor networks have been revived thanks to quantum information theory and the progress in understanding the role of entanglement in quantum many-body systems. Moreover, tensor network states have turned out to play a key role in other scientific disciplines. In this context, here I provide an overview of the basic concepts and key developments in the field. I briefly discuss the most important tensor network structures and algorithms, together with an outline of advances related to global and gauge symmetries, fermions, topological order, classification of phases, entanglement Hamiltonians, holografic duality, artificial intelligence, the 2D Hubbard model, 2D quantum antiferromagnets, conformal field theory, quantum chemistry, disordered systems and many-body localization.

Key points

  • Tensor networks are mathematical representations of quantum many-body states based on their entanglement structure.

  • Different tensor network structures describe different physical situations, such as low-energy states of gapped 1D systems, 2D systems and scale-invariant systems.

  • Variational methods over families of tensor networks enable the approximation of the low-energy properties of complex quantum Hamiltonians. Other methods also allow the simulation of time evolution, the calculation of low-energy excitations and much more.

  • Symmetric tensor network states enable more efficient simulation methods and the description of fermionic systems, lattice gauge theories, topological order and the classification of phases of quantum matter.

  • Tensor networks, such as the multiscale entanglement renormalization ansatz, have been linked to a possible lattice realization of the holographic principle in quantum gravity.

  • Tensor networks also provide a natural framework for understanding machine learning and probabilistic language models.

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Fig. 1: Diagrammatic representation of a tensor network and several examples of tensor networks.
Fig. 2: Symmetries in tensors and tensor networks.
Fig. 3: 2D projected entangled pair state (PEPS) on a cylinder.
Fig. 4: 1D MERA.

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Acknowledgements

The author acknowledged financial support from Ikerbasque, Donostia International Physics Center (DIPC) and Deutsche Forschungsgemeinschaft (DFG), as well as discussions over the years with many people on topics presented here. The author also acknowledges M. Rizzi and P. Schmoll for critical reading.

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    Román Orús

  2. Ikerbasque Foundation for Science, Bilbao, Spain

    Román Orús

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Glossary

Renormalization

The process of removing degrees of freedom that are not relevant to describe a complex system at different scales of some physical variable (energy, length…).

Anti-de Sitter

(AdS). A geometric space with negative curvature.

Conformal field theory

(CFT). A quantum field theory with conformal symmetry, which includes scale invariance. Low-energy field theories of quantum critical systems are usually CFTs.

Topological order

A type of order in quantum matter entirely due to global entanglement properties and which does not exist classically. Other characterizations: excitations are anyonic, the topological entanglement entropy is non-zero, ground states are topologically degenerate, and reduced density matrices are locally equivalent.

Area-law

Property by which the entanglement entropy of a region scales proportionally to the size of the boundary of the region.

Tensor contraction

Sum over the common indices of a set of tensors (for example, matrix multiplication).

Correlation length

Non-mathematically, this is the length scale at which correlations are sizeable in a many-body system.

Wilsonian renormalization

In this context, a renormalization scheme by which different length scales are obtained purely by removing short-distance degrees of freedom, without having previously disentangled them. In many cases, this is done by removing high-energy (and/or high-momenta) degrees of freedom.

Trotter decomposition

Decomposition of the exponential of the sum of two matrices A and B as \({{\rm{e}}}^{(A+B)t}=\mathop{{\rm{lim}}}\limits_{n\to \infty }{({{\rm{e}}}^{At/n}{{\rm{e}}}^{Bt/n})}^{n}\), with t some real parameter.

Daubechies wavelets

Family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support.

Wegner’s flow

Flow of continuous unitary transformations that diagonalizes a Hamiltonian, as \({H}_{{\rm{D}}}=\mathop{{\rm{lim}}}\limits_{l\to \infty }{U}^{\dagger }(l)HU(l)\) with l the flow parameter and HD the diagonal Hamiltonian.

Lanczos methods

An adaptation of a power method to find the eigenvalues and eigenvectors of a matrix.

Clebsch–Gordan coefficient

Coefficient of the change of basis in angular momentum, from a tensor product basis to a coupled basis (for instance, from spins \(1/2\otimes 1/2\) to spins \(0\oplus 1\).

SWAP gate

Unitary gate that swaps the quantum states of two physical systems.

Parent Hamiltonian

Hamiltonian that has a given PEPS or MPS as unique ground state.

Boltzmann machine

A specific type of neural network in which the target is to reproduce some Gibbs thermal probabilities.

MERGE

Linguistic operation introduced by Noam Chomsky, which picks up two entities (for example, noun and adjective) and produces a new one from the two (for instance, noun phrase).

Many-body localization

(MBL). Property of interacting quantum many-body systems with disorder leading to a phase of matter that does not self-thermalize.

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Orús, R. Tensor networks for complex quantum systems. Nat Rev Phys 1, 538–550 (2019). https://doi.org/10.1038/s42254-019-0086-7

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