It has been a long time since I got time to update. The last few months have been rough as well as exciting. To start off here are the project updates :
Between the last update and this, I gave my second internals at the university which went pathetic, made it to the GSoC reunion in California where I had worked on the documentation and few failing cases, and have started to read more of TQC, Hopf algebras and material on Topological Phases of Matter.
Coming to the project, the issues with the documentation had been resolved, I could not get time to revamp the code and get it working for the failing cases. Recently Miguel chipped in with the revamped code and most of the cases seem to work. The library is open for testing and it would be great to hear some comments on it.
Coming to the trip to California it was a great experience (not before a lot of travel … changed around 7 flights on the way and back … no more flights without leg space), I had an opportunity to interact with people from various other open source communities (some of them which I contributed and attempted to contribute to) and legends like Linus Torvalds, Peter Norvig. On my return,a lot of work had already been piled up and I had to catch up with various deadlines. Somehow I managed to finish them but not without the support of my friends.
Lately I have started reading through the work of Prof. Wang who has been an inspiration from the time I started to go through the topics of TQC. I have been making attempts to understand his paper titled “Quantum Computing : A Quantum Group Approach”. My understanding of it is the anyonic systems can be modeled through UMC’s (Unitary Modular Categories). The symmetries of crystals can be explained by Group Theory but not all phases of matter can be classified by this and so to classify these Topological phases of matter we use Fusion Categories (More of it can be found from the paper on String Net condensation, I had a look at it but nothing concrete). So to continue I had the understanding of Ribbon Categories (which are rigid braided monoidal categories with an additional isomophim which maps the object to the dual of the dual). This mathematical structure is in total sync with the anyonic system, as the rigidity property relates to the anti particle of the anyon, the braiding isomorphism relates to the braiding of anyons, and the monoidal structure explains how anyons can be put together and the interaction with the trivial anyon, then to explain the twist isomorphism (this can be viewed as the link formed when an anyon is braided around a trivial anyon is not same as not braiding at all) so we use the additional isomorphism (I think it can be called as a pivotal structure). So anyon can be related as an object in this category. To explain the fusion rules we require the semisimplicity and in addition to that we require the condition of modularity and that would make the system complete. This also can be related through Hopf algebras which share similar structure.
How can this be related to TQFT’s and link invariants ??
n+1 TQFT can be defined as a functor from n+1 bordism category to the category of finite dimensional vector spaces. (I just took the definition) from my understanding there can be some results which can be obtained in the 3+1 TQFT using such kind of a formulation.
Link invariants :
The axioms of RFC are used to define topological invariants of braided trivalent framed graphs in the plane. This statement is again from the paper and I am trying to still understand this.
I had the opportunity to realize why Jones polynomial computation is faster in quantum realm than in classical realm but I really cant write it down firmly I have an idea of how it works but I will make an attempt to explain it the next time.
I recently got into relating graphs and knots, and this might be clinging on for some time in the near future. I have read through Kauffman bracket formulation using graphs and its a simple enough idea on how we construct a graph out of a knot. We shade the bounded regions and mark as nodes the regions, which ever regions shares a crossing we draw an edge across it and we extend the rules.
I am again sorry if something is not clear, I am not good at writing mathematics in blogs but I hope I could provide the underlying connections and ideas.