  This paper presents a way to estimate the Heegaard genus of a $3$-manifold
using the Turaev-Viro state sum TQFT. The Turaev-Viro state sum TQFT is derived
from the modular category associated to the quantum group
$U_q(\mathfrak{sl}_2)$, which is unitary for some $q$ by Wenzl. Hence by Turaev
and Virelizier the corresponding TQFT is unitary. We modify a proof by
Garoufalidis to give a lower bound of the Heegaard genus using a unitary TQFT,
and then use the software Regina to provide some known counterexamples to the
rank versus genus conjecture.
