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Categorical Quantum Mechanics:  2017-2018



Schedule C1Computer Science

Schedule C1Mathematics and Computer Science

Schedule CMSc in Advanced Computer Science

MSc in Mathematics and Foundations of Computer Science



UPDATE: There will be a practical session on the proof assistant Globular, from 3:15pm to 5:00pm on Monday 5 March (Week 8) in room 379. See 'Practical' section below.

This course gives an introduction to the theory of monoidal categories, and investigates their application to quantum computer science. We will cover the following topics, illustrating applications throughout to quantum computation:

  • Monoidal categories, the graphical calculus, coherence
  • Linear structure on categories, biproducts, dagger-categories
  • Dual objects, traces, entangled states
  • Monoids and comonoids, copying and deleting
  • Frobenius structures, normal forms, characterizing bases
  • Complementarity, bialgebras, Hopf algebras
  • Completely positive maps, the CP construction
  • Bicategories and their graphical calculus

To complement the theoretical side of the course, we will also learn about the proof assistant Globular, and use it to formalize some of the results. There will be a practical session using the tool later in the term (date TBA).

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


Weeks 1 to 8, Mondays and Tuesdays at 2pm, Lecture Theatre A, Department of Computer Science.


The notes and slides for the entire course can be downloaded here.


There are classes in weeks 2 to 8. There are 2 groups, at 2pm on Thursday and 3pm on Friday, both in room 013 in the Robert Hook Building. Hand-in is by 9am on the preceding Tuesday, to your TA's pigeon hole in Computer Science. The classes will cover the following problems in the lecture notes:

  • Week 2: 1.4.1-1.4.5, 1.4.12.
  • Week 3: 1.4.6, 1.4.7, 2.5.1-2.5.5.
  • Week 4: 3.4.1-3.4.5, 3.4.7-3.4.9, 3.4.11
  • Week 5: 4.3.1-4.3.5, 4.3.7, 4.3.8
  • Week 6: 5.7.1-5.7.7
  • Week 7: 6.5.1-6.5.6
  • Week 8: 7.7.1-7.7.4, 7.7.7


The course will involve a practical session using the graphical reasoning package Globular. This will take place between 3:15pm and 5pm on Monday 5 March (Week 8) in room 379. Please make sure you have watched the YouTube introduction before the practical.

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 Globular software tool. 
  8. Be ready to tackle current research topics studied by the quantum group.


An ideal foundation for this course is given by the Michaelmas term course Categories, Proofs and Processes. The Hilary term course Quantum Computer Science can be taken at the same time as this course, and will cover some of the same ideas in a less mathematical way.

The necessary background for this course is 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 covers 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.


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