Ultra Low Overhead Syndrome Extraction for the Steane code

We establish a new performance benchmark for the fault-tolerant syndrome extraction of [[7, 1, 3]] Steane code with a dynamic protocol. Our method is built on two highly optimized circuits derived using fault-equivalent ZX-rewrites: a primary fault-tolerant circuit with 14 CNOTs and an efficient non-fault-tolerant recovery circuit with 11 CNOTs. The protocol uses an adaptive response to internal faults, discarding flagged measurements and falling back to the recovery circuit to correct potentially detrimental errors. Monte Carlo simulations confirm the efficiency of our protocol, reducing the logical error rate per cycle by an average of 14.3% relative to the optimized Steane method [arXiv:2506.17181] and 17.7% compared to the Reichardt's three-qubit method [arXiv:1804.06995], the leading prior techniques.