Full accuracy simulations of quantum systems are very costly, and as a result most studiesof quantum error correction assume a probabilistic Pauli error model, largely because sucherrors can be e ciently simulated. Therefore, the behaviour of more general noise in aquantum error correcting code is poorly characterized. In this thesis, we present results whichdemonstrate the scaling of the logical noise with respect to the physicaldelity, and arguethat 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 recoveryoperations appropriate to the system. We further demonstrate that when physical noiseacts on fewer than d qubits in an [[n; k; d]] code, the resultant noise is Pauli, and developa method for approximating the dominant contributions to the e ective logical noise up toa speci ed precision in terms of the physical in delity. We derive conditions under whichsets of recovery operations will produce equivalent logical noise channels, with examples ofequivalencies in the 3 qubit repetition code, the 5 qubit code, the Steane code, and the Shorcode. We also provide a general expression for the e ective logical noise when the physicalqubits undergo depolarizing or Pauli noise in a quantum error correcting code, examine thebehaviour of depolarizing noise under concatenation of the 5 qubit and Steane codes, andpresent an algorithm for soft decoding which is not subject to statistical sampling, with anemphasis on the e ective behaviour of a concatenated 5 qubit code undergoing depolarizingnoise after applying a specialized version our soft decoding algorithm.
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The Effects of Quantum Error Correction on Noisy Systems
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