Errors and pseudothresholds for incoherent and coherent noise

We compare the effect of single qubit incoherent and coherent errors on the logical error rate of the Steane [[7,1,3]] quantum error correction code by performing an exact full-density-matrix simulation of an error correction step. We find that the effective 1-qubit process matrix at the logical level reveals the key differences between the error models and provides insight into why the Pauli twirling approximation is a good approximation for incoherent errors and a poor approximation for coherent ones. Approximate channels composed of Clifford operations and Pauli measurement operators that are pessimistic at the physical level result in pessimistic error rates at the logical level. In addition, we observe that the pseudothreshold can differ by a factor of 5 depending on whether the error is calculated using the fidelity or the distance.

Figure 1

Average error rate for the ADC and its approximations for various damping strengths. The curve for the PCa at the logical level is located exactly underneath the curve for the ADC at the logical level. The level-1 pseudothresholds are given by the intersection between the best fits for the physical and the logical error magnitudes. The lines are just guides to the eye and do not correspond to the best fits.

Figure 2

Diamond distance for the ADC and its approximations for various damping strengths. The curve for the PCa at the logical level is located exactly underneath the curve for the ADC at the logical level. The lines are guides to the eye.

Figure 3

Average error rate for the RZC and its approximations for various over-rotation angles. The curves for the PCw, CMCa, and CMCw do not intersect the physical RZC curve because they all scale quadratically with $\theta$, but the coefficient of the RZC curve is the smallest. In this case, the PCw and the expanded channels provide a useless lower bound (0) to the exact pseudothreshold. The lines are guides to the eye.

Figure 4

Diamond distance for the RZC and its approximations for various over-rotation angles. The lines are guides to the eye.

Figure 5

Error magnitude for the RZC quantified with two different error metrics: the average error rate and the diamond distance. The intersection of the two blue curves gives the level-1 pseudothreshold based on the average error rate. The intersection of the two cyan curves gives the pseudothreshold based on the diamond distance. In this case, the ${\theta }_{\mathrm{th}}$ obtained from the diamond distance is about 1 order of magnitude higher than the one obtained from the average error rate, as can be seen in Table 12.

Figure 6

Normalized error magnitude for the RZC. The normalization factor is the respective error magnitude for each method at the physical level.