Achieving Algorithmically Relevant Logical Error Rates with Steane Syndrome Extraction on the Bacon–Shor Code

Previously, we showed that for low distances (d ≤ 9), Steane’s syndrome extraction method yields significant improvements in the performance of the Bacon–Shor code compared to directly measuring the stabilizers with single bare ancilla qubits [1]. Here, we extend our fault-tolerant logical state preparation protocol based on post-selected GHZ states to distances beyond nine and study the resulting scaling of logical error rates under a circuit-level depolarizing Pauli noise model. These results provide valuable insights into the code distances required for near-term fault-tolerant demonstrations as well as resource estimates needed to achieve algorithmically useful logical error rates on architectures with long-range qubit connectivity, such as trapped ions or neutral atoms.

[1] G. Escobar-Arrieta and M. Gutiérrez, Phys. Rev. A 111, 032427 (2025)