  We consider a unitary cocycle or Sch\"urmann triple on the non-commutative
unitary group fixed by a complex matrix which induces an additive free white
noise or an additive free L\'evy process on the tensor algebra over the full
Fock space. A L\'evy process on a Voiculescu dual semi-group is given by a
generator or Sch\"urmann triple. We will show how a free L\'evy process on the
non-commutative unitary group fixed by a complex matrix can be obtained by
infinitesimally convolving the additive free white noise.
