PhD Math Defence: "Degenerate Quantum Codes and Quantum Channel Capacities"

CANDIDATE:  TYLER JACKSON

No quantum system can be perfectly isolated from the environment and, as a result, no physical implementation of quantum information processing tasks can be completely free from noise. The best tool for combating such noise is the use of quantum error correcting codes (QECCs). The general requirements for QECCs has been documented for a while and yet construction of good codes and understanding their effect remains a difficult and active area of study. In this thesis I outline the work I have completed looking into both of these problems. My general technique throughout this work is to reduce the problem space as much as possible through use of group theory and then to use numerical methods and bring computational power to bear on the problem. In chapter 2, I investigate the construct of good codes for the amplitude damping channel, using the CWS framework. Through an exhaustive search method many new codes with better parameters then previously known are found. In chapter 3, I continue constructing good codes for the amplitude damping channel, this time using code concatenation techniques to find results which would be unfeasible to find via exhaustive search. Finally, chapter 4 broaches the difficult problem of determining if a channel has capacity, the ultimate use of QECCs in a sense. Developing on the key works done on the problem, and with use of computational power available today, I develop the theory surrounding effective noise channels obtained from applying a QECC on multiple uses of a channel in order to determine if capacity exists. These techniques allow us to find many very noisy non-Pauli channels that have positive capacity which previously could not be shown to have capacity.