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The Turaev-Viro invariants are scalar topological invariants ofthree-dimensional manifolds. Here we show that the problem of estimating theFibonacci version of the Turaev-Viro invariant of a mapping torus is a completeproblem for the one clean qubit complexity class (DQC1). This complements aprevious result showing that estimating the Turaev-Viro invariant for arbitrarymanifolds presented as Heegaard splittings is a complete problem for thestandard quantum computation model (BQP). We also discuss a beautiful analogybetween these results and previously known results on the computationalcomplexity of approximating the Jones polynomial.

Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit