Math 8803

Math 8803: Graduate Topics in Topology: Topological Quantum Computing/Quantum Representations

This course is designed to give students an introduction to the fields of topological quantum computing and quantum representations of braid groups (and mapping class groups) through the lens of topology.

In particular, much of this course will focus on the broad field of mathematics that has grown out of the Jones polynomial of links. Our motivation will be based on the observations of both Freedman and Kitaev that this mathematics could be applied to models of quantum computation. In a sense, we will be following the approach of Freedman-Kitaev-Larsen-Wang (seen for example in this paper) while focusing primarily on topology.

There are of course many fantastic references for each of these subjects, but I will attempt to provide only one reference that best captures that day's lecture to help keep feelings of being overwhelmed at a minimum. This is of course an imprecise science and more complete references will gladly be provided on request.

Phase 1: The Jones Representation of Braid Groups

January 20th- A Course Overview: A brief description of what this course will be and the flavor of the math that will be discussed.

January 25th- Braid Groups: An introduction to braid groups, both by presentation and as geometric braids. A great reference is the book Braid Groups by Kassel and Turaev.

January 27th- Braid Diagrams: A diagrammatic introduction to braid groups. Kassel and Turaev's book remains a great reference (in particular Chapter 1 is most relevant here).

February 1st- Algebras of Diagrams: A look at quotients of the group algebra of the braid group, including Hecke algebras and Temperley-Lieb algebras as algebras of diagrams. One great reference for this is Section 3.1 and 3.2 of "Local unitary representations of the braid group and their applications to quantum computing" by Delaney, Rowell, and Wang (which we will refer to as DRW for future references).

February 3rd- Generic Jones Representation: We define the Jones representation when the Kauffman variable A is not a root of unity. This uses the Kauffman bracket homomorphism from the braid group algebra to the Temperley-Lieb algebra. A great reference is DRW section 3.4.

February 8th- Jones-Wenzl Idempotents: We introduce particularly "nice" elements in the Temperley-Lieb algebras, and discuss their properties, and how to piece them together for admissible triples. I suggest section 2.3 of "On Picture (2+1)-TQFTS" by Freedman, Nayak, Walker, and Wang as a reference.

February 10th- Temperley-Lieb Algebra Structure: For generic values of the Kauffman variable we describe a basis of matrix units using admissibility and Jones-Wenzl idempotents. This describes the semi-simple structure of generic Temperley-Lieb algebras. A great reference for this is Section 1.1.6 of Topological Quantum Computation by Zhenghan Wang.

February 15th- Temperley-Lieb Recoupling: We see how to explicitly compute with admissibly labelled trivalent graphs. This includes changing between different bases of matrix units and evaluating the Markov pairing. The best reference for this is Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds by Kauffman and Lins, and in particular many of the details of computations that we skipped can be found here. Chapter 9 contains a very helpful reference guide that summarizes much of the topic.

February 17th- Computing the (generic) Jones representation: We return to the generic Jones representation with out basis of matrix units and the tool of recoupling in hand to compute explicit matrices for the Jones representation. I recommend Section 1.1.6 of TQC by Zhenghan Wang again as reference here.

February 22nd- The Temperley-Lieb-Jones Algebra: We look at the troubles that arise when the Kauffman variable is a root of unity. Our solution comes in the form of the Jones quotient of the Temperley-Lieb algebra. This gives a "truncated" version of the generic Temperley-Lieb algebras by only leaving "small" Jones-Wenzl projectors. I suggest section 1.2 of TQC by Zhenghan Wang for reference.

February 24th- Temperley-Lieb-Jones Algebra Structure: We use the notation of r-admissibility to describe a basis of matrix units for the Temperley-Lieb-Jones algebras when the Kauffman variable is 4rth root of unity. A discussion on dimensions of Temperley-Lieb-Jones algebras is also given. I'm again suggesting section 1.2 of TQC by Zhenghan Wang for reference.

March 1st- Ising and Fibonacci Examples: We look in depth at the Temperley-Lieb-Jones algebras at 16th roots of unity. This example is often called the Ising representation (for the correct choice of root of unity). We include a brief sketch of the finite-ness of the image of the braid group for this example. We also discuss the restriction to even labels for a 20th root of unity which is often called the Fibonacci representation. I suggest DRW section 4 and, in particular, Theorem 4.8 for the discussion of the image of the Ising braid group representation.

Phase 2: WRT Representations of Mapping Class Groups

March 3rd- The Jones Polynomial: We give definitions of links and link diagrams. Then we define the Jones polynomial using writhe and the Kauffman bracket on link diagrams. Then we discuss braid closures and see the Jones polynomial as a renormalized trace on the Jones representation. (keep in mind the "annoying distinction between Tr and tr in this context). I suggest Chapter 6.1 of The Knot Book by Colin Adams. Also, although it isn't self-contained, section 5.4 gives a great proof of Alexander's theorem.

March 8th-Dehn Surgery: We discuss the process of Dehn surgery, meaning drill and then fill along a link in the S^3. This involves introducing framed links and seeing the Lickorish-Wallace theorem which describes every closed, orientable, connected 3-manifold as the result of surgery along some framed link. I suggest chapter 9 of Knots and Links by Dale Rolfsen.

March 10th-Kirby Calculus: We look at an explicit example of Dehn surgery (on the 0 framed unknot). Then we proceed to the fundamental theorem of Kirby calculus which says that two framed links give the same 3-manifold through Dehn surgery if they are related by a finite sequence of Kirby moves. This gives us that 3-manifold invariants can be defined as framed link invariants that don't change under a finite list of moves. I recommend Chapter 12.3 of Kauffman and Lins' book for this one.

March 15th-The WRT Invariant (part 1): We look to extend/generalize the Kauffman bracket of a framed link in a way that can give invariance under Kirby moves. This involves taking a (finite) weighted sum of the Kauffman bracket of a framed link that is colored by the different Jones=Wenzl idempotents. I suggest "The skein method for three-manifold invariants" by Lickorish.

March 17th- The WRT invariant (part 2): Using the work of the previous lecture we see how our framed link invariant which is invariant under the K2 move can be renormalized to give a 3-manifold invariant (using the signature of the link). This is then generalized to an invariant of admissibly labeled ribbon graphs embedded in 3-manifolds. I suggest Chapter 13 of An Introduction to Knot Theory by Lickorish.

March 29th- The WRT Vector Space: We assign a vector space to a genus g handebody built out of r-admissibly labeled ribbon graphs in the handlebody. A focus is put on genus 1 and 2. An inner product is then constructed using #^g(S^2xS^1) and the WRT invariant which makes the basis provided orthogonal. I recommend "On central extensions of mapping class groups" by Gregor Masbaum and Justin Roberts.

March 31st- The WRT Representation: We see how the inner product of the previous lecture was really a specialization of a more general construction in the case of the identity map. By looking at gluing two genus g handlebodies together along their boundary we see how each mapping class of the boundary surface gives a linear operator on the WRT vector space again using the WRT invariant (and surgery). I recommend "Skeins and mapping class groups" by Justin Roberts.

April 5th- Dehn Twist Actions: We look at the explicit surgery diagrams used to describe the WRT action of a Dehn twist on the vector space of admissible graphs in a handlebody defined last class. This reduces understanding the projective representation to understanding 3 local pictures (modulo the projective ambiguity). I recommend the "The Reshetikhin-Turaev representation of the mapping class group" by Gretchen Wright.

April 7th- Temperley-Lieb Categories: We saw how the family of Temperley-Lieb (or Temperley-Lieb-Jones) algebras fit together into a nice family of compatible algebras called an algebra or linear category. We additionally reviewed some introductory definitions for categories. I recommend TQC by Zhenghan Wang again here.

April 12th- Fusion Categories: We look at a generalization of the TLJ categories seen last class. While this subject is a rich field of algebra our focus is on graphical calculus, and we think of morphisms as being admissibly labeled trees. I recommend Bonderson's thesis "Non-Abelian Anyons and Interferometry" for a discussion of the more concrete approach that we took while referring to Tensor Categories by Etingof, Gelaki, Nikshych, and Ostrik for the full picture.

April 14th- Heegaard Genus: We look at the lower bound on the Heegaard genus of a 3-manifold coming from quantum invariants/representations. Along the way, we discuss the lower bound coming from the rank of the fundamental group and explicit examples where this bound is not sharp. The focus is on Helen Wong's result on an example of Seifert fibered manifolds where the quantum bound is sharp while the rank is not. I recommend Helen Wong's paper "Quantum invariants can provide sharp Heegaard genus bounds".

April 19th- Quantum Computing and Computation: We discuss the abstract framework of computation and quantum computing. We keep an eye towards how what we have learned in this class can be applied in this framework. I recommend section 3 of "Mathematics of Topological Quantum Computing" by Eric Rowell and Zhenghan Wang.

April 21st- Quantum Computing and Complexity: We continue the discussion from last class, but focus on complexity classes. This leads into describing approximating the Jones polynomial as a computing problem and the BQP complexity class. I recommend Chapter 3 of TQC by Zhenghan Wang here.