  Quantum error correction protects fragile quantum information by encoding it
into a larger quantum system. These extra degrees of freedom enable the
detection and correction of errors, but also increase the operational
complexity of the encoded logical qubit. Fault-tolerant circuits contain the
spread of errors while operating the logical qubit, and are essential for
realizing error suppression in practice. While fault-tolerant design works in
principle, it has not previously been demonstrated in an error-corrected
physical system with native noise characteristics. In this work, we
experimentally demonstrate fault-tolerant preparation, measurement, rotation,
and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion
qubits. When we compare these fault-tolerant protocols to non-fault tolerant
protocols, we see significant reductions in the error rates of the logical
primitives in the presence of noise. The result of fault-tolerant design is an
average state preparation and measurement error of 0.6% and a Clifford gate
error of 0.3% after error correction. Additionally, we prepare magic states
with fidelities exceeding the distillation threshold, demonstrating all of the
key single-qubit ingredients required for universal fault-tolerant operation.
These results demonstrate that fault-tolerant circuits enable highly accurate
logical primitives in current quantum systems. With improved two-qubit gates
and the use of intermediate measurements, a stabilized logical qubit can be
achieved.
