Density Matrix Simulation of Quantum Error Correction

Sammanfattning: Quantum error correction will be integral in developing full scale quantum computers, but as of yet beneficial quantum error correction has not been demonstrated experimentally. An important question is therefore what prerequisites need to be met to achieve this. Results of density matrix simulations of the performance of the seven qubit Steane code in a quantum computing setting are presented. The full density matrix was kept throughout the whole simulations, which means that all errors can be accounted for. In particular, the importance of the circuit depth, i.e. the number of gates in series before error correction is applied, for the overall performance was investigated. It was found that the depth of the circuit has a large impact on the threshold error rate for which error correction becomes beneficial. A gain parameter was defined, which describes the largest constant factor by which errors can be suppressed. It was shown that there is a trade-off between the threshold error and the gain; The highest threshold value was found to be around p_th~10^-4, which is in line with other estimates, but the maximum gain for this value was only 3. To achieve a gain of 100, an error rate of p_err~2x10^-9 is required. In addition, performance statistics such as run-time as a function of circuit depth and width for the matlab code used for simulations are presented.