  The Turaev-Viro invariants are scalar topological invariants of
three-dimensional manifolds. Here we show that the problem of estimating the
Fibonacci version of the Turaev-Viro invariant of a mapping torus is a complete
problem for the one clean qubit complexity class (DQC1). This complements a
previous result showing that estimating the Turaev-Viro invariant for arbitrary
manifolds presented as Heegaard splittings is a complete problem for the
standard quantum computation model (BQP). We also discuss a beautiful analogy
between these results and previously known results on the computational
complexity of approximating the Jones polynomial.
