The Effects of Quantum Error Correction on Noisy Systems

Abstract

Full accuracy simulations of quantum systems are very costly, and as a result most studies of quantum error correction assume a probabilistic Pauli error model, largely because such errors can be e ciently simulated. Therefore, the behaviour of more general noise in a quantum error correcting code is poorly characterized. In this thesis, we present results which demonstrate the scaling of the logical noise with respect to the physical delity, and argue that the e ective logical noise approaches a Pauli channel as the code distance increases, even when no recovery operations are applied. As a result, we argue that the average logical delity can be used to accurately quantify the e ective logical noise, and to select recovery operations appropriate to the system. We further demonstrate that when physical noise acts on fewer than d qubits in an [[n; k; d]] code, the resultant noise is Pauli, and develop a method for approximating the dominant contributions to the e ective logical noise up to a speci ed precision in terms of the physical in delity. We derive conditions under which sets of recovery operations will produce equivalent logical noise channels, with examples of equivalencies in the 3 qubit repetition code, the 5 qubit code, the Steane code, and the Shor code. We also provide a general expression for the e ective logical noise when the physical qubits undergo depolarizing or Pauli noise in a quantum error correcting code, examine the behaviour of depolarizing noise under concatenation of the 5 qubit and Steane codes, and present an algorithm for soft decoding which is not subject to statistical sampling, with an emphasis on the e ective behaviour of a concatenated 5 qubit code undergoing depolarizing noise after applying a specialized version our soft decoding algorithm.