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Abstract:

Abstract Continuing our exploration of the detailed structure of anyon theories, we explain how one can describe the same Hilbert space with different basis sets. Changing between these bases is done with the so-called F-matrix which is a gauge dependent quantity. The F-matrices satisfy a set of consistency equations known as the pentagon equations which are “rigid” up to Gauge transformations (i.e., solutions are discrete and cannot be deformed small amounts to produce other solutions). We again use Fibonacci and Ising anyons as simple examples.

Abstract:

Abstract We give a brief introduction to Chern-Simons theory starting with Abelian Chern-Simons theory and making the connection to the charge-flux model. We discuss the vacuum partition function and the idea of a manifold invariant. We then move on to discuss Nonabelian Chern-Simons theory and the idea of Wilson loop operations, and Witten’s famous connection between knot invariants and Chern-Simons theory. The Chern-Simons theory is a paradigm for topological quantum field theories or TQFTs.

Abstract:

Abstract We introduce some basic ideas of quantum computing including the idea of a quantum circuit model and the idea of a universal quantum computer — implementing arbitrary gates arbitrarily accurately. We discuss how braiding of certain anyons can be used as a universal quantum computer, and we illustrate this using Fibonacci anyons as an example. We then turn to the task of designing braids to perform particular computations, which is known as Topological Quantum compiling.

Abstract:

Abstract Atiyah systematized the general principles of topological quantum field theories in terms of the mathematics of gluing together manifolds and combining vector spaces. A TQFT roughly is a partition function Z(M) with M a manifold, that returns a number of M has no boundary and a wavefunction on the boundary if M has a boundary. Manifolds can be glued together on their boundaries by taking inner products of wavefunctions. We roughly describe this approach and describe how it relates to the physics of anyons as we have discussed in prior chapters. We also introduce the important notion of the modular S-matrix as a change of basis.

Abstract:

Abstract The Doubled-Semion model is another simple example of generalizing the toric code. Here we start with a planar diagram algebra that is the d=-1, Z2 loop gas. I.e, here a loop is given the value of -1. This is also known as the nontrivial cocyle of Z2. This model can be thought of as the simplest case of a twisted Kitaev Quantum Double, and it is also the simplest example of the Drinfeld double of a modular anyon theory (the double of the semion theory). In this case we construct a Hamiltonian such again we have a loop gas, but now flipping over a plaqutte incurs a sign in the wavefunction if we change the parity of the number of loops. We again turn to the tube algebra to identify the excitations of the model, which are both right- and left-handed semions. We run into a problem related to the nontrivial Frobenius-Schur indicator of the input diagram algebra. However, here we can resolve the issue by choosing a convenient gauge.