  We demonstrate that the two inequivalent spinor representations of the braid
group \B_{2n+2}, describing the exchanges of 2n+2 non-Abelian Ising anyons in
the Pfaffian topological quantum computer, are equivalent from computational
point of view, i.e., the sets of topologically protected quantum gates that
could be implemented in both cases by braiding exactly coincide. We give the
explicit matrices generating almost all braidings in the spinor representations
of the 2n+2 Ising anyons, as well as important recurrence relations. Our
detailed analysis allows us to understand better the physical difference
between the two inequivalent representations and to propose a process that
could determine the type of representation for any concrete physical
realization of the Pfaffian quantum computer.
