Hello quantum formalists,
Just a quick reminder that the registration for lecture 7 is now open via https://www.crowdcast.io/e/topology-crash-course-7.
Now, a community member recently asked me this interesting question on Discord:
What is the connection between differential geometry and quantum computing?
My answer was the following:
For our purposes, in a nutshell, Lie Groups are the connection between the main parts of Differential Geometry that we're covering in the crash course and quantum computing. This is because the quantum gates form a Lie Group structure i.e. both the unitary group U(n) and the special unitary SU(n) are Lie Groups (Compact Lie Groups). This is an important technical detail that introductory quantum computing courses normally don't mention because it is a topic that is more suitable to advanced graduate/research level folks!
Now, Lie Groups don't just carry an algebraic structure (group structure). They do also carry a 'smooth manifold' structure which is part of Differential Geometry. Hence, before you learn what a Lie Group is, it helps if you know what a smooth manifold is. This is exactly the motivation of this crash course!
On a side note, one of my social media connections recently shared a lecture from the Qiskit QML Summer School about Quantum Kernels. I had a quick scan through it, and I can tell you that it has the structure of Lie Groups behind the scenes when the lecturer was playing with G=SU(2) as an example!
You will be able to see this once we cover Lie Groups in Module II. I also recommend you to see the applications of Lie Groups in Machine Learning via this paper entitled 'Survey on Lie Group Machine Learning'.
Anyway, I think it’s becoming more clear that the mathematics that we’ll be covering in the upcoming courses will not just be relevant to physics and quantum computing. Machine learning, geometric deep learning in particular, is another area where the stuff that you will learn is useful.
Finally, I highly encourage you to join our Discord community server https://discord.gg/SPcmcsXMD2.
See you this coming Friday at 5 pm BST!