  Steane's 7-qubit quantum error-correcting code admits a set of fault-tolerant
gates that generate the Clifford group, which in itself is not universal for
quantum computation. The 15-qubit Reed-Muller code also does not admit a
universal fault-tolerant gate set but possesses fault-tolerant T and
control-control-Z gates. Combined with the Clifford group, either of these two
gates generate a universal set. Here, we combine these two features by
demonstrating how to fault-tolerantly convert between these two codes,
providing a new method to realize universal fault-tolerant quantum computation.
One interpretation of our result is that both codes correspond to the same
subsystem code in different gauges. Our scheme extends to the entire family of
quantum Reed-Muller codes.
