Topological aspects of two-dimensional quantum systems

In part I, we describe a protected qubit which is realized in a two-dimensional array of Josephson junctions. Our construction is the magnetic analogue of (‘dual’ to) a suggestion of a superconducting current mirror qubit (Kitaev,2006b). Our proposal therefore inherits the intrinsic fault-tolerance of the current mirror qubit, but may perform better than it in the laboratory, since magnetic noise is generally less of a problem than electric noise. We adapt the scheme for universal fault-tolerant quantum computation proposed by Kitaev to our construction. In part II, we describe a method of detecting the Chern number and entanglement properties of topological four-component free-fermion systems in cold atom experiments. We show that the Chern number of these systems decomposes into a sum of subsystem winding numbers which can be measured from time-of-flight images. Such images also enable the degree of subsystem entanglement in, and the component entanglement spectra of, these systems to be measured. The method is applied to the quantum spin-Hall insulator and a staggered topological superconductor. We find that the phase diagrams are accurately reproduced, except when the subsystems are highly entangled.