D. J. Myers, H. Sati, U. Schreiber. Topological Quantum Gates in Homotopy Type Theory (2023)

Link to the paper: https://arxiv.org/abs/2303.02382

For my current research project, I quite rashly promised that I would find a way to apply type theory to quantum computing, and the funding body agreed with me. I worked with Bart Jacobs a few years ago on a similar project (see Nijmegen Quantum Logic Group) and the work we did looked promising but did not fully reach a satisfactory system.

I was very excited when Urs Schreiber's group's latest project was brought to my attention by my PhD student Lorenzo Perticone (see link at the top of this post). They claim to have found a way to use homotopy type theory as a language for describing quantum theory, including topological quantum gates and error correction. This would be a great achievement if true, but I will need to do a lot of background reading before I can understand the system.

The Background section of the paper is very impressive - a comprehensive reading list on both the physics and mathematics behind quantum computing, algebraic topology, higher category theory, and everything else I've been meaning to learn about for a long time.

So I thought I might as well work my way through the list, and make my notes public as I go. I plan to alternate between the physics and the mathematics references. I don't know how far I'll get or if anyone is going to be interested in this journey, but let's get started.

The first item on the list is Yuri Manin's paper Computable and Uncomputable, where the idea of a quantum computer first appeared.

Next: Y. I. Manin, Computable and Uncomputable, Sov. Radio (1980), published in: Mathematics as Metaphor: Selected essays of Yuri I. Manin, Collected Works 20, AMS (2007), 69-77, [ISBN:978-0-8218-4331-4].

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