Introduction to Topological Quantum Computing/Quantum Representations

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Not regularly scheduled

Special topics course on Topological Quantum Computing/Quantum Representations, offered in Spring 2021 by Wade Bloomquist.


An effort will be made to make this course accessible to as broad of an audience as possible. The following topics could be considered a rough list of prerequisite concepts:Topology of surfaces (for example isotopy of curves); Group theory; Linear algebra

Course Text: 
  • Topological Quantum Computation by Wang. (book available for free). 
  • Tensor Categories by Etingof, Gelaki, Nikshych, and Ostrik (book available for free).
  • A finiteness property for braided fusion categories by Naidu and Rowell (paper) 
  • Various other papers for specific results.


Topic Outline: 

This course will provide an introduction to topological quantum computation from the viewpoint of quantum representations of braid groups and mapping class groups. In short, we will look at how these groups coming from topology can act on vector spaces of diagrams that encode algebraic data. Motivated by viewing these actions as the building blocks of quantum circuits, the Property F Conjecture provides hope for understanding the general structure of these representations. The main goal of this course will be to state and understand the Property F Conjecture and its generalizations.


• Temperley-Lieb algebras/category 

• Jones representation of braid groups 

• Jones polynomial and quantum circuits 

• Introduction to tensor categories 

• Braided fusion categories/braid group representations from tensor categories 

• Property F Conjecture 

• Modular tensor categories 

• Quantum representations of mapping class groups and quantum invariants of links 

• The genus g Property F Conjecture.