Categories for Quantum Theory

This is a PhD course for the Spring 2020 organised by Rasmus Ejlers Møgelberg, Radu-Cristian Curticapean, and Michael Kastoryano. Interested students should contact Rasmus for further information on the course, including how to register.

Description

With the future prospect of quantum computers, quantum theory is increasingly an area of interest among computer scientists. This course studies quantum theory from the perspective of category theory, assuming no previous knowledge in either area. We study monoids, Frobenious structure and Hopf algebras as well as the ZX calculus, a sound and complete graphical calculus for quantum computation.

The course has been approved for 5ECTS credit by the ITU PhD council. To obtain this, students must present at least once at the seminar, and have 3 mandatory exercises approved.

Course material

We use the book Categories for Quantum Theory by Chris Heunen and Jamie Vicary.

Supplementary material:

• a note on quantum teleportation by Robin Kaarsgaard

Schedule

We will meet every Wednesday 12-14 in Auditorium 1 at ITU starting March 11, 2020. The tentative schedule is as follows.

• March 11 (Rasmus and Michael): Introduction, basics of category theory, hilbert spaces and quantum information
• March 18 (Christian): Monoidal Categories
• March 25 (Robin): Linear structure
• April 1 (Andrea): Dual objects
• April 15 (Mikkel): Monoids and comonoids
• April 22 (Magnus): Frobenious structure (sec 5.1-5.3)
• April 29 (Rasmus): Frobenious structure (sec 5.4-5.6)
• May 6 (Radu): Complementarity
• May 13 (Robin): Complete positivity
• May 20 (Severin): Complete positivity (ctd)
• May 27: Monoidal 2-categories

Mandatory exercises

• Exercise 1, due April 17: Do the following exercises from the book: 1.6, 2.2, and 3.3.
• Exercise 2, due May 10: Do the following exercises from the book: 4.3, 5.3 and 5.7. In 5.7.(b,c) describe the disconnected dagger Frobenius structures.
• Exercise 3, due June 2: Do the following from the book: 6.1, 7.3, 7.10.