We point out some major technical and conceptual mistakes which invalidate the conclusion drawn in "Anyonic braiding in optical lattices" by C. Zhang, V. W. Scarola, S. Tewari, and S. Das Sarma published in PNAS 104, 18415 (2007).
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 $\ensuremath{\pi}∕8$ gate, are also explicitly implemented by quasihole braidings. Instead of the $\ensuremath{\pi}∕8$ 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 g3.