  We study the density of the Burau representation from the perspective of a
non-semisimple TQFT at a fourth root of unity. This gives a TQFT construction
of Squier's Hermitian form on the Burau representation with possibly mixed
signature. We prove that the image of the braid group in the space of possibly
indefinite unitary representations is dense. We also argue for the potential
applications of non-semisimple TQFTs toward topological quantum computation.
