Fault-tolerant quantum computation with asymmetric Bacon-Shor codes

Bacon-Shor codes are quantum subsystem codes which are constructed by combining together two quantum repetition codes, one protecting against $Z$ (phase) errors and the other protecting against $X$ (bit flip) errors. In many situations, for example flux qubits, the noise is biased such that faults that produce $Z$ errors are much more common than faults that produce $X$ errors; in these cases it is natural to consider an asymmetric Bacon-Shor code where the code protecting against $Z$ errors is longer than the code protecting against $X$ errors. This work describes fault-tolerant constructions for gadgets that achieve universal fault-tolerant quantum computation using asymmetric Bacon-Shor codes. Gadgets take advantage of the Bacon-Shor structure by breaking up into parallel smaller gadgets that act on a single row or column, with majority voting of the separate results. For a bias of $ \epsilon/\epsilon' = 10^{4}$, we prove a threshold around $2.5 \times 10^{-3}$. The effective error strength is shown to decrease rapidly (faster than polynomial) with decreasing $\epsilon$. Therefore it may be practical to use Bacon-Shor codes directly with no additional concatenation. This could greatly reduce the resource overhead required for fault-tolerant computation with biased noise.