Realization of real-time fault-tolerant quantum error correction

Correcting errors in real-time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, mid-circuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a ten qubit QCCD (quantum charge-coupled device) trapped-ion quantum computer to encode a single logical qubit using the color code. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical SPAM error of 1.7(2) 10^{-3}$, compared to the average physical SPAM error 2.4(4) 10^{-3} of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections, which are tracked software or applied physically.