The Effects of Quantum Error Correction on Noisy Systems
Stefanie J. Beale
Full accuracy simulations of quantum systems are very costly, and as a result most studies \nof quantum error correction assume a probabilistic Pauli error model, largely because such \nerrors can be e ciently simulated. Therefore, the behaviour of more general noise in a \nquantum error correcting code is poorly characterized. In this thesis, we present results which \ndemonstrate the scaling of the logical noise with respect to the physical delity, and argue \nthat the e ective logical noise approaches a Pauli channel as the code distance increases, \neven when no recovery operations are applied. As a result, we argue that the average logical \n delity can be used to accurately quantify the e ective logical noise, and to select recovery \noperations appropriate to the system. We further demonstrate that when physical noise \nacts on fewer than d qubits in an [[n; k; d]] code, the resultant noise is Pauli, and develop \na method for approximating the dominant contributions to the e ective logical noise up to \na speci ed precision in terms of the physical in delity. We derive conditions under which \nsets of recovery operations will produce equivalent logical noise channels, with examples of \nequivalencies in the 3 qubit repetition code, the 5 qubit code, the Steane code, and the Shor \ncode. We also provide a general expression for the e ective logical noise when the physical \nqubits undergo depolarizing or Pauli noise in a quantum error correcting code, examine the \nbehaviour of depolarizing noise under concatenation of the 5 qubit and Steane codes, and \npresent an algorithm for soft decoding which is not subject to statistical sampling, with an \nemphasis on the e ective behaviour of a concatenated 5 qubit code undergoing depolarizing \nnoise after applying a specialized version our soft decoding algorithm.
•Aug 23, 2018•0 citations