  Quantum error correction (QEC) is an essential step towards realising
scalable quantum computers. Theoretically, it is possible to achieve
arbitrarily long protection of quantum information from corruption due to
decoherence or imperfect controls, so long as the error rate is below a
threshold value. The two-dimensional surface code (SC) is a fault-tolerant
error correction protocol} that has garnered considerable attention for actual
physical implementations, due to relatively high error thresholds ~1%, and
restriction to planar lattices with nearest-neighbour interactions. Here we
show a necessary element for SC error correction: high-fidelity parity
detection of two code qubits via measurement of a third syndrome qubit. The
experiment is performed on a sub-section of the SC lattice with three
superconducting transmon qubits, in which two independent outer code qubits are
joined to a central syndrome qubit via two linking bus resonators. With
all-microwave high-fidelity single- and two-qubit nearest-neighbour entangling
gates, we demonstrate entanglement distributed across the entire sub-section by
generating a three-qubit Greenberger-Horne-Zeilinger (GHZ) state with fidelity
~94%. Then, via high-fidelity measurement of the syndrome qubit, we
deterministically entangle the otherwise un-coupled outer code qubits, in
either an even or odd parity Bell state, conditioned on the syndrome state.
Finally, to fully characterize this parity readout, we develop a new
measurement tomography protocol to obtain a fidelity metric (90% and 91%). Our
results reveal a straightforward path for expanding superconducting circuits
towards larger networks for the SC and eventually a primitive logical qubit
implementation.
