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Lectures on Quantum Tensor Networks

Jacob Biamonte

TL;DR

This work surveys tensor networks as a unifying graphical language bridging computer science, quantum physics, and mathematics. It develops the Penrose graphical calculus, wire-bending dualities, and a diagrammatic SVD to factor and reason about states, operators, and processes on equal footing. The book then applies these tools to Matrix Product States and Boolean Tensor Networks, illustrating both theoretical structures (Schmidt decompositions, invariants) and practical aspects (MPS factorizations, sampling, and software resources). Together, it provides a comprehensive framework for diagrammatic reasoning in quantum information and beyond, with a toolkit spanning foundations, representations, and computational implementations.

Abstract

Situated as a language between computer science, quantum physics and mathematics, tensor network theory has steadily grown in popularity and can now be found in applications ranging across the entire field of quantum information processing. This book aims to present the best contemporary practices in the use of tensor networks as a reasoning tool, placing quantum states, operators and processes on the same compositional footing. The book has 7 parts and over 40 subsections which took shape in over a decade of teaching. In addition to covering the foundations, the book covers important applications such as matrix product states, open quantum systems and entanglement $-$ all cast into the diagrammatic tensor network language. The intended audience includes those in quantum information science wishing to learn about tensor networks. It includes scientists who have employed tensor networks in their modeling codes who have interest in the tools graphical reasoning capacity. The audience further includes the graduate student researcher, whom with some effort, should find this book accessible. I would appreciate it if you emailed me about any mistakes or typos you find.

Lectures on Quantum Tensor Networks

TL;DR

This work surveys tensor networks as a unifying graphical language bridging computer science, quantum physics, and mathematics. It develops the Penrose graphical calculus, wire-bending dualities, and a diagrammatic SVD to factor and reason about states, operators, and processes on equal footing. The book then applies these tools to Matrix Product States and Boolean Tensor Networks, illustrating both theoretical structures (Schmidt decompositions, invariants) and practical aspects (MPS factorizations, sampling, and software resources). Together, it provides a comprehensive framework for diagrammatic reasoning in quantum information and beyond, with a toolkit spanning foundations, representations, and computational implementations.

Abstract

Situated as a language between computer science, quantum physics and mathematics, tensor network theory has steadily grown in popularity and can now be found in applications ranging across the entire field of quantum information processing. This book aims to present the best contemporary practices in the use of tensor networks as a reasoning tool, placing quantum states, operators and processes on the same compositional footing. The book has 7 parts and over 40 subsections which took shape in over a decade of teaching. In addition to covering the foundations, the book covers important applications such as matrix product states, open quantum systems and entanglement all cast into the diagrammatic tensor network language. The intended audience includes those in quantum information science wishing to learn about tensor networks. It includes scientists who have employed tensor networks in their modeling codes who have interest in the tools graphical reasoning capacity. The audience further includes the graduate student researcher, whom with some effort, should find this book accessible. I would appreciate it if you emailed me about any mistakes or typos you find.

Paper Structure

This paper contains 89 sections, 49 theorems, 475 equations, 16 figures, 5 tables.

Table of Contents

  1. From Tensors to Networks
  2. Penrose Graphical Notation for Tensor Networks
  3. Tensor Network
  4. Penrose Wire Bending Duality
  5. Bending and exchanging wires.
  6. Equivalence class.
  7. Bending Density Operator Wires
  8. Bipartite Matrix Operations
  9. Vectorization of Matrices
  10. Worked Examples
  11. Problems
  12. Further Reading
  13. Matrix Product States
  14. The Diagrammatic SVD
  15. Matrix Product Factorization of States
  16. ...and 74 more sections

Key Result

Lemma 2.1

Given a tensor $T^{ij}$ with fixed labels $i,j$ we can use cups and caps to arrive at the $\text{\sf SWAP}$ operation reorders $i$ and $j$ and then the cups and caps yield In general, for a tensor with a total of $n$ indices, each index can be up or down, yielding $2^n$ possibilities. The symmetry group formed by $\text{\sf SWAP}$ is of order $n!$ and acts to arrange the $n$ legs of a tensor, yi

Figures (16)

  • Figure 1: (Diagrammatic summary of steps I, II and III). The quantum state (a) is iteratively factored into the 1D Matrix Product State (d). This procedure readily extends to $n$-body states.
  • Figure 2: Conversion from our notation (a), to conventional MPS notation (b). The factorization methods we have reviewed here allow one to "zoom in" and expose internal degree of freedom (a) or "zoom out" and expose high-level structure (b). The equational representation of the MPS in (b) is given in \ref{['eqn:MPS']}.
  • Figure 3: Example of the Boolean quantum $\text{\sf AND}$-state or tensor from CTNS. In (a) the tensors output is contracted with $\bra{1}$ resulting in the tensor splitting to the product state $\ket{11}$. In (b) the tensors output is contracted with $\bra{0}$ resulting in the entangled state $\ket{00}+\ket{01}+\ket{10}$.
  • Figure 4: A general Boolean quantum state arising from function $f$ can either be formed as (a) by network contraction with a logical-one at the output of the circuit as described by \ref{['eqn:Booleaninput']} or (b) by bending the output of the tensor network around, as in \ref{['eqn:Booleanstates']}.
  • Figure 5: Diagrammatic equations satisfied by a fixed point pair (see Definition \ref{['def:fixedpoints']}).
  • ...and 11 more figures

Theorems & Definitions (269)

  • Definition 1.1
  • Remark
  • Remark
  • Remark
  • Remark : Diagram convention---top to bottom, or right to left
  • Remark
  • Remark
  • Example : The $\epsilon$ tensor
  • Example : Concurrence and entanglement
  • Example : Quantum circuits
  • ...and 259 more