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 p_X and the phase error probability p_Z, and determine how the probability of a logical error depends on p_X and p_Z. 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 (p_Z / p_X >> 1) and the physical error rate p_Z 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.