  We extend the topological quantum computation scheme using the Pfaffian
quantum Hall state, which has been recently proposed by Das Sarma et al., in a
way that might potentially allow for the topologically protected construction
of a universal set of quantum gates. We construct, for the first time, a
topologically protected Controlled-NOT gate which is entirely based on
quasihole braidings of Pfaffian qubits. All single-qubit gates, except for the
pi/8 gate, are also explicitly implemented by quasihole braidings. Instead of
the pi/8 gate we try to construct a topologically protected Toffoli gate, in
terms of the Controlled-phase gate and CNOT or by a braid-group based
Controlled-Controlled-Z precursor. We also give a topologically protected
realization of the Bravyi-Kitaev two-qubit gate g_3.
