  $\pi/8$ phase gates (magic gates or T-gates) are crucial to augment
topological systems based on Majorana zero modes to full quantum universality.
We present a scheme based on a combination of projective measurements and
non-adiabatic evolution that effectively cancels smooth control errors when
implementing phase gates in Majorana-based systems. Previous schemes based on
adiabatic evolution are susceptible to problems arising from small but finite
dynamical phases that are generically present in topologically unprotected
gates. A measurement-only approach eliminates dynamical phases. For
non-protected gates, however, forced-measurement schemes are no longer
effective which leads to low success probabilities of obtaining the right
succession of measurement outcomes in a measurement-only implementation. We
show how to obtain a viable measurement-based scheme which dramatically
increases the success probabilities by evolving the system non-adiabatically
with respect to the degenerate subspace in between measurements. We outline
practical applications of our scheme in recently proposed quantum computing
designs based on Majorana tetrons and hexons.
