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

Roman Orus

TL;DR

The paper provides a practical, beginner-friendly overview of tensor network methods, focusing on MPS for 1D and PEPS for 2D systems and their numerical applications. It explains how TNs encode entanglement efficiently, introduces the diagrammatic language, and outlines core techniques for computing expectation values and finding ground states via variational optimization and imaginary-time evolution. Key contributions include clear descriptions of canonical forms, environment-based contractions, and stability considerations, framing TNs as powerful tools that target the area-law-restricted corner of the Hilbert space. The discussion emphasizes both the strengths and limitations of MPS/PEPS in capturing gapped and critical behavior, and it highlights practical algorithms and representations that underpin modern simulations of quantum many-body systems.

Abstract

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 Entangled Pair States (PEPS). Selected details on some of the associated numerical methods for 1d and 2d quantum lattice systems are also discussed.

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

TL;DR

The paper provides a practical, beginner-friendly overview of tensor network methods, focusing on MPS for 1D and PEPS for 2D systems and their numerical applications. It explains how TNs encode entanglement efficiently, introduces the diagrammatic language, and outlines core techniques for computing expectation values and finding ground states via variational optimization and imaginary-time evolution. Key contributions include clear descriptions of canonical forms, environment-based contractions, and stability considerations, framing TNs as powerful tools that target the area-law-restricted corner of the Hilbert space. The discussion emphasizes both the strengths and limitations of MPS/PEPS in capturing gapped and critical behavior, and it highlights practical algorithms and representations that underpin modern simulations of quantum many-body systems.

Abstract

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 Entangled Pair States (PEPS). Selected details on some of the associated numerical methods for 1d and 2d quantum lattice systems are also discussed.

Paper Structure

This paper contains 27 sections, 59 equations, 45 figures.

Table of Contents

  1. Introduction
  2. A bit of background
  3. Why Tensor Networks?
  4. New boundaries for classical simulations
  5. New language for (condensed matter) physics
  6. Entanglement induces geometry
  7. Hilbert space is far too large
  8. Tensor Network theory
  9. Tensors, tensor networks, and tensor network diagrams
  10. Breaking the wave-function into small pieces
  11. MPS and PEPS: generalities
  12. Matrix Product States (MPS)
  13. Some properties
  14. Some examples
  15. Projected Entangled Pair States (PEPS)
  16. ...and 12 more sections

Figures (45)

  • Figure 1: (color online) (a) The DNA is the fundamental building block of a person. In the same way, (b) the tensor is the fundamental building block of a quantum state (here we use a diagrammatic notation for tensors that will be made more precise later on). Therefore, we could say that the tensor is the DNA of the wave-function, in the sense that the whole wave-function can be reconstructed from it just by following some simple rules.
  • Figure 2: (color online) Two examples of tensor network diagrams: (a) Matrix Product State (MPS) for 4 sites with open boundary conditions; (b) Projected Entangled Pair State (PEPS) for a $3 \times 3$ lattice with open boundary conditions.
  • Figure 3: (color online) The entanglement entropy between $A$ and $B$ scales like the size of the boundary $\partial A$ between the two regions, hence $S \sim \partial A$.
  • Figure 4: (color online) The manifold of quantum states in the Hilbert space that obeys the area-law scaling for the entanglement entropy corresponds to a tiny corner in the overall huge space.
  • Figure 5: (color online) Tensor network diagrams: (a) scalar, (b) vector, (c) matrix and (d) rank-3 tensor
  • ...and 40 more figures