Optimal Bacon-Shor codes
John C. Napp, John P. Preskill · Quantum Information and Computation · 2013
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 ($p_Z/p_X\gg 1)$ 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.