# A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States

@article{Ors2013API, title={A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States}, author={Rom{\'a}n Or{\'u}s}, journal={Annals of Physics}, year={2013}, volume={349}, pages={117-158} }

This is a partly non-technical introduction to selected topics on tensor network methods, based on several lectures and introductory seminars given on the subject. It should be a good place for newcomers to get familiarized with some of the key ideas in the field, specially regarding the numerics. After a very general introduction we motivate the concept of tensor network and provide several examples. We then move on to explain some basics about Matrix Product States (MPS) and Projected…

## Figures and Topics from this paper

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## 1,052 Citations

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## References

SHOWING 1-10 OF 219 REFERENCES

Implementing global Abelian symmetries in projected entangled-pair state algorithms

- Computer Science, Physics
- 2011

A formalism to implement Abelian symmetries in two-dimensional tensor-network states is explained and benchmark results are presented that confirm the validity of these approximations in the context of projected entangled-pair state algorithms.

Simulating strongly correlated quantum systems with tree tensor networks

- Physics
- 2010

We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the…

Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems

- Physics
- 2012

In this paper we explore the practical use of the corner transfer matrix and its higher-dimensional generalization, the corner tensor, to develop tensor network algorithms for the classical…

Tensor Network States and Geometry

- Mathematics, Physics
- 2011

Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different…

Computational complexity of projected entangled pair states.

- Physics, MedicinePhysical review letters
- 2007

It is shown how PEPS can be used to approximate ground states of gapped Hamiltonians and that creating them is easier than creating arbitrary PEPS, and how the latter two tasks are both proven to be #P-complete.

Time evolution of projected entangled pair states in the single-layer picture

- Physics
- 2011

We propose an efficient algorithm for simulating quantum many-body systems in two spatial dimensions using projected entangled pair states. This is done by approximating the environment, arising in…

Tensor network states and algorithms in the presence of a global SU(2) symmetry

- Physics
- 2012

The benefits of exploiting the presence of symmetries in tensor network algorithms have been extensively demonstrated in the context of matrix product states (MPSs). These include the ability to…

Holographic geometries of one-dimensional gapped quantum systems from tensor network states

- Physics
- 2012

A bstractWe investigate a recent conjecture connecting the AdS/CFT correspondence and entanglement renormalization tensor network states (MERA). The proposal interprets the tensor connectivity of the…

Renormalization of tensor-network states

- Physics
- 2010

We have discussed the tensor-network representation of classical statistical or interacting quantum lattice models, and given a comprehensive introduction to the numerical methods we recently…

Non-abelian symmetries in tensor networks: A quantum symmetry space approach

- Physics
- 2012

Abstract A general framework for non-abelian symmetries is presented for matrix-product and tensor-network states in the presence of well-defined orthonormal local as well as effective basis sets.…