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Abstract:

Abstract TQFTs, or anyon theories, have a large amount of addition mathematical structure which we explore in this chapter. We show that the quantum dimension which defines the weight of a loop in the diagram algebra is the same as the quantum dimension that describes the size of the many-anyon Hilbert space. We define the S-matrix in terms of diagrams, and the T-matrix in terms of twists. We show how the S- and T-matrices are related to the modular group of diffeomorphisms of the torus, and related further to the notion of central charge. We discuss how the many constraints on TQFTs enables one to develop a table of all possible sufficiently “small” TQFTs given certain very general conditions. We give tables for both modular and super-modular theories. We introduce the Kirby Color or Omega Strand as an important tool.

Abstract:

Abstract In this chapter we begin the detailed structure of anyon theories, or TQFTs. We introduce the concept of fusion and explain how it is related to the Hilbert space of the anyon systems. We emphasize the importance of locality as a principle for understanding anyons. Associativity is a strong constraint on possible particle fusion rules. As examples, we introduce both Fibonacci anyons and Ising anyons. We explain how the fusion rules of an anyon theory can be used to calculate the degeneracy of the ground state on arbitrary 2-manifolds.

Abstract:

Abstract We briefly introduce some basic ideas of information and quantum information. We start by comparing a classical memory with a quantum memory. Simple repetition codes can protect classical memories from error and we introduce the idea of a code space, physical bits, and logical bits. However, the quantum no-cloning theorem prevents a straightforward generalization to the quantum case. Nonetheless quantum error correction is indeed possible. We show the 9 qubit Shor code as an example.

Abstract:

Abstract We introduce the toric code as an example of a quantum error correcting code. The system is a lattice of spins on a torus. We use a basis of spin up/down spins which we translate into diagrams of colored (or uncolored) edges of the lattice. We introduce commuting vertex and plaquette operators, and explain that particular eigenstates of these operators define our code-space which in the diagram language corresponds to sums over loop configuration. The code space decomposes into four orthogonal wavefunctions which are distinguished by the parity of the number of colored loops around each of the two cycles of the torus. Quantum information is stored in the coefficients of these four wavefunctions in superposition. We then turn to consider error processes. We discover that errors occur in pairs locally and are moved apart from each other by “strings” of additional error operators. We discover that it is easy to identify when physical errors have occurred and correct them before a logical error occurs. We extend the discussion to consider irregular “lattices” of spins and surfaces with arbitrary genus (and make the connection to the Shor Code). Finally we turn to consider the ZN generalization of the toric code.