Logical-qubit operations in an error-detecting surface code

Future fault-tolerant quantum computers will require storing and processing data in logical qubits. Here we realize a suite of operations on distance-2 surface code qubit built from seven physical qubits stabilized using repeated error-detection cycles. Logical include initialization into arbitrary states, measurement the cardinal bases Bloch sphere universal set single-qubit gates. For each type operation, observe higher performance for variants over non-fault-tolerant variants, quantify difference. In particular, demonstrate process tomography gates, notion Pauli transfer matrix. This integration high-fidelity with scalable scheme stabilization is milestone road to error correction higher-distance superconducting codes. Large-scale manipulate information encoded error-corrected A complete has now been realized detection.