Quantum architecture search for syndrome extraction in fault-tolerant quantum computation

Quantum error correction (QEC) is a key enabler of fault-tolerant quantum computing, yet the strong code-level properties of modern quantum low-density parity-check (qLDPC) codes do not automatically translate into equally strong circuit-level performance. A central bottleneck lies in syndrome extraction, where CNOT ordering, global scheduling, and ancilla faults jointly determine error propagation and logical reliability. We introduce syndrome-extraction architecture (SEA) search, which jointly designs local CNOT orderings, global $X/Z$ scheduling, and selective flag deployment. Focusing on Abelian two-block group-algebra (2BGA) codes, we exploit their group structure to share ordering templates across stabilizer measurements while retaining flexible flag placement. We develop a probabilistic quantum architecture search framework that samples candidate architectures, compiles them into valid global circuits using CP-SAT, and updates the architecture distribution from decoder-in-the-loop circuit-level feedback. Across 20 Abelian 2BGA codes, SEA search consistently achieves the lowest logical error rate among all baselines, including solver-based scheduling, fully flagged scheduling, and AlphaSyndrome. Moreover, the resulting circuits recover the code-level distance in 16 of the 20 evaluated instances. These results show that syndrome extraction is an architecture-level design problem and establish quantum architecture search as a practical methodology for fault-tolerant QEC circuit design.