Tensors in Quantum Information Theory

This is the webpage for the doctoral course Tensors in Quantum Information Theory, given in spring 2025 at the LPT Toulouse by Ion Nechita.

Schedule

There will be 10 lecturs of 2h each, taking place in the room 20, ground floor of the 3R1 building, as follows:

N.	Date	Topic
1.	Mon, March 17th, 2pm-4pm	Meet and greet, introduction to tensors, first examples, basics of random matrix theory
2.	Wed, March 19th, 2pm-4pm	Tensors. Graphical notation
3.	Mon, March 24th, 2pm-4pm	Graphical notation in applications. Contracting tensor networks.
4.	Wed, March 26th, 2pm-4pm	The Singular Value Decomposition
5.	Mon, March 31st, 2pm-4pm	Tensor rank
6.	Wed, April 2nd, 2pm-4pm	Tensor norms
7.	Mon, April 7th, 2pm-4pm	Project choice
8.	Wed, April 9th, 2pm-4pm	Project work
9.	Mon, April 14th, 2pm-4pm	Project work
10.	Wed, April 16th, 2pm-4pm	Project presentations

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Topics

1. Basics of tensors
2. Graphical notation
3. Tensor ranks
4. Tensors in quantum information theory
5. Tensor norms
6. Tensor decompositions
7. Tensor eigenvalues

Projects

1. Computational complexity

• NP hardness of tensor problems
• Undecidable tensor problems

2. Quantum Information Theory

• Tensor norms for non local games
• Tensor norms for entanglement
• Tensor norms for quantum incompatibility and quantum steering
• Tensor eigenvalues
• Algorithms for injective and projective norms

3. Tensor networks

• Contracting tensor networks
• TensorKrowch https://github.com/joserapa98/tensorkrowch
• Chapters 3- from Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks

4. ML and NN using tensors

• AlphaTensor
• Tensor PCA
• The transformer architecture

Lecture notes

... available as we go, either pdfs or notebooks ...

Here are the notes of a series of 3 lectures I gave in February about similar topics: ICTS 2025 - Tensor norms for quantum entanglement.

Lecture 1

We discussed some problems in random matrix theory where the notion of rank is important.

We started with the Netflix problem, aka matrix completion. The Netflix problem tackles the challenge of predicting missing ratings in a large, sparse user - item matrix by leveraging the assumption that the underlying data is low rank. This low rank assumption implies that there exist only a few latent factors - such as genre preferences, viewing habits, or intrinsic qualities of movies and shows - that largely determine how users rate content. In essence, even though the observable matrix is incomplete and noisy, it is believed that user preferences and item attributes can be effectively captured by a small number of these hidden factors. This compressed representation allows matrix factorization methods to accurately reconstruct the full matrix, making it possible to provide reliable recommendations even from limited data.

We then discussed the basics of Random Matrix Theory, see the notebook.

Lecture 2

Handwritten notes.

Formal introduction to tensor products, tensors, tensor contraction. Penrose graphical notation for tensors. Applications.

References for the graphical formalism:

• R. Penrose - Applications of negative dimensional tensors
• R. Penrose - The road to reality -> Chapter 12.8
• J. Bridgeman, C. Chubb - Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• J. Biamonte, V. Bergholm - Tensor Networks in a Nutshell
• J. Taylor - Graphical tensor notation for interpretability -> discusses Transformers

Lecture 3

Handwritten notes.

Continue discussing and applying the graphical formalism. Vectorization. The computational cost of contracting tenor networks: see Section 1.4 here and, e.g. here for the important notion of treewidth.

Lecture 4

Handwritten notes.

The Singular Value Decomposition (here's a refresher on the SVD).

Lecture 5

Handwritten notes, see also the ICST lecture notes.

The tensor rank is a notion of fundamental importance in the study of tensors. A good reference for QIT related questions is the paper Tensor rank and entanglement of pure quantum states by Bruzda, Friedland, and Życzkowski.

Lecture 6

Handwritten notes, see also the ICST lecture notes.

Lectures 7-8-9

Project work on the three selected topics:

• TensorKrowch for ground states
• The transformer architecture
• Spectra of random tensor networks

Lecture 10

The students presented the projects, here are some relevant links to their amazing work:

• TensorKrowch for ground states: GitHub repository containing slides for the presentation as well as python and Mathematica notebooks.
• The transformer architecture: the students presented the main ingredients of the transformer architecture (in particular the attention mechanism) and provided an example of anomaly detection with and without attention.
• Spectra of random tensor networks: GitHub repository containing a jupyter notebook for visualizing the singular value distribution of flattenings of tensor network states defined using TensorKrowch.