An accurate analytical framework for computing fault-tolerance thresholds using the [[7,1,3]] quantum code

Abstract

In studies of the threshold for fault-tolerant quantum error-correction, it is generally assumed that the noise channel at all levels of error-correction is the depolarizing channel. The effects of this assumption on the threshold result are unknown. We address this problem by calculating the effective noise channel at all levels of error-correction specifically for the Steane [[7,1,3]] code, and we recalculate the threshold using the new noise channels. We present a detailed analytical framework for these calculations and run numerical simulations for comparison. We find that only X and Z failures occur with significant probability in the effective noise channel at higher levels of error-correction. We calculate that when changes in the noise channel are accounted for, the value of the threshold for the Steane [[7,1,3]] code increases by about 30 percent, from .00030 to .00039, when memory failures occur with one tenth the probability of all other failures. Furthermore, our analytical model provides a framework for calculating thresholds for systems where the initial noise channel is very different from the depolarizing channel, such as is the case for ion trap quantum computation.