Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders

Quantum error correction offers a promising path for performing high fidelity quantum computations. Although fully fault-tolerant executions of algorithms remain unrealized, recent improvements in control electronics and quantum hardware enable increasingly advanced demonstrations of the necessary operations for error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. We encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow for the correction of any single fault in the circuitry. Using real-time feedback, we reset syndrome and flag qubits conditionally after each syndrome extraction cycle. We report decoder dependent logical error, with average logical error per syndrome measurement in Z(X)-basis of ~0.040 (~0.088) and ~0.037 (~0.087) for matching and maximum likelihood decoders, respectively, on leakage post-selected data.