  Quantum topological invariants have played an important role in computational
topology, and they are at the heart of major modern mathematical conjectures.
In this article, we study the experimental problem of computing large $r$
values of Turaev-Viro invariants $\mathrm{TV}_r$. We base our approach on an
optimized backtracking algorithm, consisting of enumerating combinatorial data
on a triangulation of a 3-manifold. We design an easily computable parameter to
estimate the complexity of the enumeration space, based on lattice point
counting in polytopes, and show experimentally its accuracy. We apply this
parameter to a preprocessing strategy on the triangulation, and combine it with
multi-precision arithmetics in order to compute the Turaev-Viro invariants. We
finally study the improvements brought by these optimizations compared to
state-of-the-art implementations, and verify experimentally Chen and Yang's
volume conjecture on a census of closed 3-manifolds.
