Categorical Quantum Mechanics: 2014-2015
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Term | Hilary Term 2015 (16 lectures) |
Overview
Note about the exam: in past years the exam for this course has been quite hard. We will ensure this year that the exam is of a more ordinary difficulty, comparable with other advanced MFoCS courses offered by the department.
This course gives an introduction to some advanced topics in category theory, and shows how we can use them to model phenomena in quantum computer science. We will aim to cover the following topics:
- 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
- The Quantomatic proof assistant
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. Lecture notes are available for download via the 'Course materials' link in the right-hand menu.
Lectures
Lectures will take place in weeks 1 to 8, on Mondays and Thursdays. All lectures will be at 10am in the Tony Hoare room.
Classes
There are exercise classes in weeks 2 to 8, covering the following problems:
- Week 2: 1.4.1, 1.4.2, 1.4.4, 1.4.5, 1.4.6
- Week 3: 1.4.7-1.4.9, 2.5.1-2.5.5
- Week 4: 3.3.1-3.3.5, 3.3.7-3.3.9
- Week 5: 4.5.1-4.5.5, 4.5.7, 4.5.8
- Week 6: 5.7.1-5.7.4, 5.7.6, 5.7.8, 5.7.9
- Week 7: 6.5.1--6.5.6
Practical
The course will involve a practical session using the graphical reasoning package Quantomatic, on Tuesday 2-4pm of week 8, in Computer Room 379.
Learning outcomes
After studying this course, students will be able to:
- Understand and prove basic results about monoidal categories.
- Fluently manipulate the graphical calculus for compact categories.
- Model quantum protocols categorically and prove their correctness graphically.
- Appreciate differences between categories modeling classical and quantum theory.
- Work with Frobenius algebras in monoidal categories.
- Manipulate quantum algorithms in the ZX-calculus.
- Explore graphical theories using the quantomatic software tool.
- Be ready to tackle current research topics studied by the quantum group.
Prerequisites
Ideal foundations for this course are given by the Michaelmas term courses Categories, Proofs and Processes and 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.
Synopsis
Reading list
The course materials will be made available each week before the lectures. Please do report all typos, errors, or other suggestions for improvement!
Related research
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Feedback
Students are formally asked for feedback at the end of the course. Students can also submit feedback at any point here. Feedback received here will go to the Head of Academic Administration, and will be dealt with confidentially when being passed on further. All feedback is welcome.
Taking our courses
This form is not to be used by students studying for a degree in the Department of Computer Science, or for Visiting Students who are registered for Computer Science courses
Other matriculated University of Oxford students who are interested in taking this, or other, courses in the Department of Computer Science, must complete this online form by 17.00 on Friday of 0th week of term in which the course is taught. Late requests, and requests sent by email, will not be considered. All requests must be approved by the relevant Computer Science departmental committee and can only be submitted using this form.