  We implement logically encoded three-qubit circuits for the quantum Fourier
transform (QFT), using the [[7,1,3]] Steane code, and benchmark the circuits on
the Quantinuum H2-1 trapped-ion quantum computer. The circuits require multiple
logical two-qubit gates, which are implemented transversally, as well as
logical non-Clifford single-qubit rotations, which are performed by
non-fault-tolerant state preparation followed by a teleportation gadget. First,
we benchmark individual logical components using randomized benchmarking for
the logical two-qubit gate, and a Ramsey-type experiment for the logical $T$
gate. We then implement the full QFT circuit, using two different methods for
performing a logical control-$T$, and benchmark the circuits by applying it to
each basis state in a set of bases that is sufficient to lower bound the
process fidelity. We compare the logical QFT benchmark results to predictions
based on the logical component benchmarks.
