[Paper Review] Tensor Networks in a Nutshell

Jacob Biamonte, Ville Bergholm|arXiv (Cornell University)|Jul 31, 2017

Quantum many-body systemsPhysics and Astronomy52 references121 citations

TL;DR

This tutorial introduces tensor networks and their diagrammatic language, explains how to represent quantum states with matrix product states and related networks, and shows how tensor contractions solve counting problems and relate to quantum circuits.

ABSTRACT

Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and pictorially reason about quantum circuits, channels, protocols, open systems and more. Our goal is to explain tensor networks and some associated methods as quickly and as painlessly as possible. Beginning with the key definitions, the graphical tensor network language is presented through examples. We then provide an introduction to matrix product states. We conclude the tutorial with tensor contractions evaluating combinatorial counting problems. The first one counts the number of solutions for Boolean formulae, whereas the second is Penrose's tensor contraction algorithm, returning the number of $3$-edge-colorings of $3$-regular planar graphs.

Motivation & Objective

• Introduce the graphical tensor network language and its historical context.
• Explain the connection between tensor networks and quantum circuits via diagrammatic notation.
• Present matrix product states and their role in efficiently representing certain quantum states.
• Demonstrate tensor contractions as tools for solving combinatorial counting problems.
• Highlight key tensor operations such as bending wires, cups, caps, and the SVD within the diagrammatic framework.

Proposed method

• Present tensors as labeled shapes with open legs and illustrate contractions as wires.
• Use diagrammatic rules to perform operations like bending wires (cups/caps) and crossings (SWAP).
• Introduce the diagrammatic SVD and its role in obtaining Schmidt decompositions and MPS representations.
• Show how to build matrix product states by iteratively applying SVD across bipartitions and grouping results.
• Demonstrate how tensor networks can count combinatorial problems via contractions, including Penrose’s contraction techniques.

Experimental results

Research questions

• RQ1 How can tensor networks be used to visually and algebraically represent quantum states, circuits, and contractions?
• RQ2 What is the role of the diagrammatic SVD in constructing efficient state representations like MPS?
• RQ3 How do diagrammatic manipulations (cups, caps, SWAP) relate to standard linear-algebra operations and invariants?
• RQ4 In what ways can tensor contractions count combinatorial objects and connect to known algorithms like Penrose’s contraction?
• RQ5 How do entanglement properties relate to the topology and Schmidt coefficients in tensor-network representations?

Key findings

• Tensor networks provide an efficient approximate representation of certain quantum states through regular structures and contraction schemes.
• Matrix product states enable compact 1D state representations when entanglement is limited, with cost scaling linearly in the number of parties given bounded bond dimension.
• The diagrammatic SVD yields Schmidt decompositions and underpins MPS construction and truncation via the Eckart–Young–Mirsky principle.
• Bending and crossing wires (cups, caps, SWAP) give rise to index transformations and map–state dualities that unify states and operators.
• The epsilon/tensor tools and CNOT/COPY/XOR relations illustrate how common quantum gates can be realized and analyzed in a tensor-network language.
• Diagrammatic techniques connect quantum circuits, entanglement invariants, and state representations, enabling both conceptual and computational insights.