Majorana anyons, non-Abelian statistics and quantum computation in Chern–Simons–Higgs theory

Abstract

We naturally obtain the NOT and CNOT logic gates, which are key pieces of quantum computing algorithms, in the framework of the non-Abelian Chern–Simons–Higgs theory in two spatial dimensions. For that, we consider the anyonic quantum vortex topological excitations and show that Majorana anyons, namely, self-adjoint combinations of these vortices and anti-vortices, have in general non-Abelian statistics. The associated unitary monodromy braiding matrices become the required logic gates in the special case when the vortex spin is s = 1/4, which corresponds to the case of Ising non-Abelian anyons, found in different quantum computing systems. We explicitly construct the vortex field operators, show that they carry both magnetic flux and charge and obtain their Euclidean correlation functions by using the method of quantization of topological excitations, which is based on the order-disorder duality. These correlators are in general multivalued, the number of sheets being determined by the vortex spin. This, by its turn, is proportional to the vacuum expectation value of the Higgs field and therefore can be tuned by both the free parameters of the Higgs potential and the temperature.