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

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

2011-05-25

arxiv.org/abs/1105.5100v2

Physics

Quantum Physics

16 pages, 8 figures, presented at TQC '11. Added references

Scientific paper

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.

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