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Categorical Quantum Mechanics:  2012-2013


Teaching Assistant


MSc in Mathematics and Foundations of Computer Science



Category theory gives a powerful mathematical framework for working with quantum theory, and provides a high-level computer science perspective with which to understand it. This course gives an overview of some of the recent research in this exciting field, much of which was carried out here at the Department of Computer Science. The focus is on reformulating quantum-mechanical concepts in category-theoretical terms, and applying this approach to quantum foundations and quantum information. The categorical formalism has a pictorial representation which makes deductions intuitive, and this will form a major part of the course.

While we concentrate on the applications to quantum theory, category theory forms an enormously important part of the modern mathematical landscape, and the tools we introduce in this course have close relationships to other area of mathematics, including representation theory, quantum field theory and knot theory. They also have relevance to more applied disciplines, such as programming language semantics and computational linguistics. Further topics may be investigated if time permits.

This course can currently only be taken by students enrolled on the DPhil or MSc in MFoCS programmes. However, everyone is welcome to sit in and follow the lectures.

Lecture notes are available for download via the "Course materials" link in the right-hand menu.


Exercises for week 1: 1.6.1, 1.6.2. (Hand in by Wednesday of week 1, 6pm.)
Exercises for week 2: 1.6.4-1.6.7. (Hand in by Wednesday of week 2, 6pm.)
Exercises for week 3: 1.6.3, 2.5.1-2.5.5. (Hand in by Wednesday of week 3, 6pm.)
Exercises for week 4: 2.5.6-2.5.9, 3.7.1-3.7.2. (Hand in by Thursday of week 5, 6pm.)
Exercises for week 5: 3.7.3-3.7.9. (Hand in by Thursday of week 5, 6pm.)


The course will involve a practical session using the graphical reasoning package “Quantomatic”. This will be held in week 8. Date, time, and place to be confirmed.

Learning outcomes

After studying this course, students will be able to:

  1. Understand and prove basic results about monoidal categories. 
  2. Fluently manipulate the graphical calculus for compact categories. 
  3. Model quantum protocols categorically and prove their correctness graphically. 
  4. Appreciate differences between categories modeling classical and quantum theory. 
  5. Work with Frobenius algebras in monoidal categories. 
  6. Manipulate quantum algorithms in the ZX-calculus. 
  7. Explore graphical theories using the quantomatic software tool. 
  8. Be ready to tackle current research topics studied by the quantum group.


Ideal foundations for this course are given by the Michaelmas term course ''Categories, Proofs and Processes'', and the Hilary term course ''Quantum Computer Science''. Students who have not taken these courses will need to be familiar with basic topics from category theory and linear algebra, including categories, functors, natural transformations, vector spaces, Hilbert spaces and the tensor product. Chapter zero in the lecture notes briefly recall this background material.

Students wishing to do their dissertation with the Quantum Group are expected to sit this course, as well as the two mentioned above.


The lecture notes cover the following topics. Depending on the lectures, they may not all be discussed.

  • Symmetric monoidal categories
  • Graphical calculus
  • Duals for morphisms
  • Duals for objects
  • Copying and deleting
  • Frobenius algebras and classical structures
  • Modelling quantum protocols
  • Categories of completely positive maps
  • Complementary observables
  • Axiomatizing entangled states

Reading list

The lecture notes can be found under "Course Materials" on the right.

Related research at the Department of Computer Science