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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.

Abstract:

Abstract How Lord Kelvin and Peter Tait tried to describe atoms with topology and created the field of knot theory.

Abstract:

Abstract We introduce the idea of a knot invariant using the Kauffman bracket as an example and explore a few key ideas of knot theory. We then make the relationship to physical topological quantum systems whose dynamics are determined by knot invariants. We show the relationship to quantum mechanics and briefly argue that quantum computation can be done with such topological quantum systems. We mention some of the experimental systems that are topological, particularly including fractional quantum Hall effect.