Comparing Shor and Steane error correction using the Bacon-Shor code

Quantum states decohere through interaction with the environment. Quantum error correction can preserve coherence through active feedback wherein quantum information is encoded into a logical state with a high degree of symmetry. Perturbations are detected by measuring the symmetries of the state and corrected by applying gates based on these measurements. To measure the symmetries without perturbing the data, ancillary quantum states are required. Shor error correction uses a separate quantum state for the measurement of each symmetry. Steane error correction maps the perturbations onto a logical ancilla qubit, which is then measured to check several symmetries simultaneously. We experimentally compare Shor and Steane correction of bit flip errors using the Bacon-Shor code implemented in a chain of 23 trapped atomic ions. We find that the Steane method produces fewer errors after a single round of error correction and less disturbance to the data qubits without error correction.