  We introduce a Floquet circuit describing the driven Ising chain with
topological defects. The corresponding gates include a defect that flips spins
as well as the duality defect that explicitly implements the Kramers-Wannier
duality transformation. The Floquet unitary evolution operator commutes with
such defects, but the duality defect is not unitary, as it projects out half
the states. We give two applications of these defects. One is to analyze the
return amplitudes in the presence of "space-like" defects stretching around the
system. We verify explicitly that the return amplitudes are in agreement with
the fusion rules of the defects. The second application is to study unitary
evolution in the presence of "time-like" defects that implement anti-periodic
and duality-twisted boundary conditions. We show that a single unpaired
localized Majorana zero mode appears in the latter case. We explicitly
construct this operator, which acts as a symmetry of this Floquet circuit. We
also present analytic expressions for the entanglement entropy after a single
time step for a system of a few sites, for all of the above defect
configurations.
