Improved performance of the Bacon-Shor code with Steane's syndrome extraction method

We compare Steane's and Shor's syndrome extraction methods on the Bacon-Shor code. We propose a straightforward strategy based on postselection to prepare the logical ${|0\ensuremath{\rangle}}_{L}$ and ${|+\ensuremath{\rangle}}_{L}$ states of the Bacon-Shor code by using flaglike qubits to verify their constituent Greenberger-Horne-Zeilinger states. We perform stabilizer simulations with a circuit-level, depolarizing Pauli error model and find that Steane's method significantly outperforms Shor's. Not only does Steane's method result in pseudothresholds that are about one order of magnitude higher than Shor's, but also its advantage increases monotonically as we go from a distance-3 to a distance-9 Bacon-Shor code. The advantage of Steane's method is the greatest in the regime where gate errors dominate over measurement errors. Some of the circuit constructions we propose for Steane's method are not formally fault tolerant yet outperform the formally fault-tolerant Shor's protocols for experimentally relevant physical error rates. This suggests that constructing formally fault-tolerant circuits that maintain the full code distance is not strictly necessary to guarantee the usefulness of a quantum error-correcting protocol. Despite relying on postselection, we find that our methods can be efficient. These protocols would be naturally implementable on a platform with long-range qubit interactions like trapped ions or neutral atoms.