Realizing Topological Quantum Computation through Category Theory

[Amit]

Hello ,
          I am Amit. I would like to discuss the implementation of modular tensor categories for realization of Topological Quantum Computation. In Topological Quantum Computation we use anyon braiding (which develops a phase) to construct quantum gates which are nothing but unitary transforms. The entire process of anyon braiding can be mathematically modeled by representation of modular tensor categories. I have attached the article through which I have gone through for understanding the implementation in CAS. In addition to what is already present, most of the work is on matrices. The 2 Vect spaces are isomorphic to whole numbers, 1 cells are represented by matrices and 2 cells are represented by inner matrices (or 2 matrices) i.e., matrices inside matrices. Most of the implementation is by using the properties of matrices and extending the present implementation of matrices. The implementation then tested the Fibonacci model by mentioning the fusion rules. I would like add this functionality to Sympy and if possible carry it out as a part of GSoC 2014. According to my understanding the implementation is mostly mathematical but the physical extension is to TQC by testing the Fibonacci model. I request the community to comment on this. Thanks.

File : http://ora.ox.ac.uk/objects/uuid%3Ac9b6eaf8-29d4-4637-a576-5a35d3c957bb/datastreams/THESIS01

[Amit Jamadagni]

It would be really helpful if someone could discuss on the feasibility of such kind of an idea. Thanks.

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[Joachim Durchholz]

Am 24.02.2014 09:12, schrieb Amit Jamadagni:
> It would be really helpful if someone could discuss on the feasibility of
> such kind of an idea. Thanks.

Hm. Well. SymPy is a volunteer project, and all participants have other
important and/or interesting things to do. Since your question requires
specialist knowledge, the specific persons who can answer it will have
to schedule time for it.

>>
http://ora.ox.ac.uk/objects/uuid%3Ac9b6eaf8-29d4-4637-a576-5a35d3c957bb/datastreams/THESIS01

Also, reading a 100-page thesis and doing a check what it would take to
make SymPy deal with the math in it isn't something that can be done
within a mere 13 hours, not even by somebody who grabs the issue instantly.
Besides, most of us live in Europe or America, so most of us have been
sleeping.

In general, SymPy does have symbolic computations for vector math, and I
suppose tensors are supported as well.
I don't know whether SymPy can make use of the isomorphism with whole
numbers, but then math isn't my strong point, so somebody else will have
to comment on that.

Regards,
Jo

[Brian Granger]

Amit,

Thanks for your interest. While I am very much interested in seeing
the quantum computing capabilities of SymPy improve, I think this
particular proposal is out of scope for the project.

* The material is advanced enough that there is no one of the SymPy
team that would be capable to mentoring this project. I am probably
the closest (Physics professor, focusing on quantum mechanics), but I
have absolutely no background in this stuff. This point is extremely
important because we have found that perhaps the most important
ingredient for a successful GSoC experience is an active mentor who
has a deep understanding of the material.
* The material is specialized enough that I don't think it belongs in
SymPy, even if we had a mentor for it. Honestly, even the general
quantum computing stuff already borders on being too specialized for
SymPy. Were I to do it again today, I would probably make the quantum
computing stuff a separate package. Oh, well.

Given these two factors, I think the best path forward for you is to
develop a separate package for topological quantum computing that uses
SymPy, but is separate. That is one of the great things about SymPy -
you can easily extend it in separate projects/packages. I know that
does't help you get GSoC funding for this though :(

Cheers,

Brian

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Brian E. Granger
Cal Poly State University, San Luis Obispo
bgra...@calpoly.edu and elli...@gmail.com

[Amit Jamadagni]

Hello Granger,
          Thanks for the reply. It was just an idea and as per my understanding most of it was mathematical rather than physics, with the extension going into physics (In sense representing the process using mathematics) and this model was extended to test Fibonacci model. Anyways I have ideas for quantum error correction and I have done some study from Nielsen and Chuang regarding Pauli groups and their applications to stabilizers. I have browsed through the issue posted on issue tracker but I could not understand the code that well. Is there a possibility of extending the quantum module in this direction.If so what are things that are to be worked on?? Thanks. 

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