Quantum Codes From Symmetry

Abstract

The Eastin-Knill theorem shows that the transversal gates of a quantum code, whichare naturally fault-tolerant, form a finite group G. We show that G is an invariant of equivalent quantum codes and thus can be considered as a well defined symmetry. This thesis studies how the symmetry G dictates the existence and parameters of quantum codes using representation theory. We focus on qubit quantum codes that have symmetry coming from finite subgroups ofSU(2). We examine two different methods of deriving quantum codes from these symmetries. The first method is concrete but not very general, it only applies when G is a binary dihedral subgroup BDa of SU(2). The second method is abstract but more general. Not only does it apply to all subgroups of SU(2), but it highlights the role that symmetry plays in logical errors and unveils a hidden time-reversal symmetry. From each of these methods we produce many examples of novel qubit code families.