  We realize a suite of logical operations on a distance-two logical qubit
stabilized using repeated error detection cycles. Logical operations include
initialization into arbitrary states, measurement in the cardinal bases of the
Bloch sphere, and a universal set of single-qubit gates. For each type of
operation, we observe higher performance for fault-tolerant variants over
non-fault-tolerant variants, and quantify the difference through detailed
characterization. In particular, we demonstrate process tomography of logical
gates, using the notion of a logical Pauli transfer matrix. This integration of
high-fidelity logical operations with a scalable scheme for repeated
stabilization is a milestone on the road to quantum error correction with
higher-distance superconducting surface codes.
