Optimal Bacon-Shor codes

TL;DR

This paper analyzes the performance of Bacon-Shor quantum codes, identifying optimal block sizes and demonstrating their effectiveness against biased noise and low physical error rates, with bounds on logical error rates under noisy syndrome data.

Contribution

It determines optimal block sizes for Bacon-Shor codes under independent noise and shows their effectiveness against biased noise without concatenation.

Findings

Single Bacon-Shor codes protect well against biased noise

Optimal block size depends on error probabilities p_X and p_Z

Logical error rate bounds are derived for noisy syndrome data

Abstract

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.