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

TL;DR

This paper demonstrates that Steane's syndrome extraction method significantly improves the performance of the Bacon-Shor code over Shor's method, especially in regimes with dominant gate errors, by using post-selection and flag qubits.

Contribution

The authors propose a post-selection strategy with flag qubits for preparing Bacon-Shor code states and show Steane's method outperforms Shor's in simulations, even with circuits that are not fully fault-tolerant.

Findings

Steane's method yields higher pseudo-thresholds than Shor's.

The advantage of Steane's method increases with code distance.

Some non-fault-tolerant circuits outperform fault-tolerant Shor's protocols.

Abstract

We compare Steane's and Shor's syndrome extraction methods on the Bacon-Shor code. We propose a straightforward strategy based on post-selection to prepare the logical $|0\rangle_L$ and $|+\rangle_L$ states of the Bacon-Shor code by using flag-like qubits to verify their constituent Greenberger-Horne-Zeilinger states. We perform stabilizer simulations with a depolarizing Pauli error model and find that Steane's method significantly outperforms Shor's. Not only does Steane's method result in pseudo-thresholds that are about 1 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 suggest 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 post-selection, 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.