Optimal Bacon-Shor codes

We study the performance of Bacon-Shor codes, quantum subsystem codes which are well suited for applications to fault-tolerant quantum memory because the error syndrome can be extracted by performing two-qubit measurements. Assuming independent noise, we find the optimal block size in terms of the bit-flip error probability pX and the phase error probability pZ, and determine how the probability of a logical error depends on pX and pZ. We show that a single Bacon-Shor code block, used by itself without concatenation, can provide very effective protection against logical errors if the noise is highly biased (pZ/pX 1) and the physical error rate pZ is a few percent or below. We also derive an upper bound on the logical error rate for the case where the syndrome data is noisy.