It is an oft-cited fact that no quantum code can support a set of
fault-tolerant logical gates that is both universal and transversal. This no-go
theorem is generally responsible for the interest in alternative universality
constructions including magic state distillation. Widely overlooked, however,
is the possibility of non-transversal, yet still fault-tolerant, gates that
work directly on small quantum codes. Here we demonstrate precisely the
existence of such gates. In particular, we show how the limits of
non-transversality can be overcome by performing rounds of intermediate
error-correction to create logical gates on stabilizer codes that use no
ancillas other than those required for syndrome measurement. Moreover, the
logical gates we construct, the most prominent examples being Toffoli and
controlled-controlled-Z, often complete universal gate sets on their codes. We
detail such universal constructions for the smallest quantum codes, the 5-qubit
and 7-qubit codes, and then proceed to generalize the approach. One remarkable
result of this generalization is that any nondegenerate stabilizer code with a
complete set of fault-tolerant single-qubit Clifford gates has a universal set
of fault-tolerant gates. Another is the interaction of logical qubits across
different stabilizer codes, which, for instance, implies a broadly applicable
method of code switching.Comment: 18 pages + 5 pages appendix, 12 figure
