Error Correction of Quantum Channels

Abstract

As manifested in Shor's groundbreaking seminal work, quantum mechanics promise the possibility of having substantially more effective computation devices. This is in fact the result of quantum parallelism: the coherent interference pattern between the multitude of superpositions. But the fragility of a quantum state, which on one hand is used to take advantage of the power of entanglement, also can result in undesired interference between the state of the quantum system, carrying or storing the information, with the environment. The problem of maintaining quantum coherence remains one of the most important obstacles in the attempt of exploiting the new possibilities opened up by applications of quantum mechanics in classical computations. Redundancy, which is the main tool in all classical error correcting schemes is no more available in quantum setting, because of the no-cloning theorem, that prevents the duplication of quantum states. The seminal independent work of Shor [32] and Steane [35] gave birth to the current active theory of quantum error correction, which is the subject of this thesis. This thesis is a non-technical introduction to the theory of quantum error correction. We present the mathematical formalism for the theory, covering the main ideas of correcting schemes and techniques, laid in a mathematical framework, avoiding unnecessary computational and physical technicalities. Foremost, I would like to thank my supervisors Dr. Mehrdad Kalantar and Dr. Matthias Neufang for their priceless guidance, support and encouragement provided throughout my graduate studies. I am more than thankful to these wonderful professors for their time and effort devoted during these two years. I would also like to thank Dr. Kennedy for his kind support and help during these years. I also want to thank the School of Mathematics and Statistics and their kind staff. Finally, I would like to thank my lovely parents, my dearest sister and my amazing friends, especially Pegah, for their unconditional love and support. I am very lucky to have these beautiful people during this journey.