Information amidst noise: preserved codes, error correction, and fault tolerance in a quantum world

Abstract

Quantum coherence is the key ingredient for characteristically quantum effects. It allows for radically different technologies than those using classical systems, including quantum communication, quantum computation, and other devices with quantum control. Quantum coherences are, however, extremely fragile and susceptible to damage from environmental noise. The success of any experiment or technology based on quantum phenomena demands careful preservation of quantum coherences within the system. The study of the effects of noise on a quantum system, and how to prevent loss of coherence is the central theme of this thesis. Starting from basic principles behind how information is stored in a system and what it means for it to be preserved, we build up a framework that allows one to understand what kind of information can survive through a noise process. The resulting elegant matrix-algebraic description of information-preserving structures within a quantum system characterizes codes that can perfectly preserve information in the presence of noise. Our framework encompasses examples like pointer states, noiseless subsystems and error-correcting codes. Furthermore, it leads to a simple, analytical approach to approximate quantum error correction. While perfect quantum error correction is a standard method used to protect information from noise, approximate error correction allows for the use of a smaller quantum system to store the same information, without sacrificing much in resilience against noise. Asking what happens to information stored in a quantum system when the encoding and recovery procedures in error correction are also noisy leads to the concept of fault tolerance. Fault tolerance provides schemes, built upon quantum error correction, that enable accurate simulation of a quantum computation even when the elementary gates are imperfect. Realistic gates used to build a fault-tolerant circuit, however, often require additional noise-suppression techniques in order for any quantum effects to be observed at all. A common technique is dynamical decoupling. We demonstrate how dynamical decoupling in elementary gates can be rigorously accounted for in the fault-tolerance analysis, and show how, under the right conditions, it can lead to fault-tolerant circuits with less stringent noise and resource requirements.