  A non-Clifford gate is required for universal quantum computation, and,
typically, this is the most error-prone and resource intensive logical
operation on an error-correcting code. Small, single-qubit rotations are
popular choices for this non-Clifford gate, but certain three-qubit gates, such
as Toffoli or controlled-controlled-Z (CCZ), are equivalent options that are
also more suited for implementing some quantum algorithms, for instance those
with coherent classical subroutines. Here, we calculate error rates and
resource overheads for implementing logical CCZ with pieceable fault-tolerance,
a non-transversal method for implementing logical gates. We provide a
comparison with a non-local magic-state scheme on a concatenated code and a
local magic-state scheme on the surface code. We find the pieceable
fault-tolerance scheme particularly advantaged over magic states on
concatenated codes and in certain regimes over magic states on the surface
code. Our results suggest that pieceable fault-tolerance is a promising
candidate for fault-tolerance in a near-future quantum computer.
