Estimating TuraevViro threemanifold invariants is universal for quantum computation
Abstract
The TuraevViro invariants are scalar topological invariants of compact, orientable 3manifolds. We give a quantum algorithm for additively approximating TuraevViro invariants of a manifold presented by a Heegaard splitting. The algorithm is motivated by the relationship between topological quantum computers and (2+1)dimensional topological quantum field theories. Its accuracy is shown to be nontrivial, as the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer. Thus approximating certain TuraevViro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation. This establishes a relation between the task of distinguishing nonhomeomorphic 3manifolds and the power of a general quantum computer.
 Publication:

Physical Review A
 Pub Date:
 October 2010
 DOI:
 10.1103/PhysRevA.82.040302
 arXiv:
 arXiv:1003.0923
 Bibcode:
 2010PhRvA..82d0302A
 Keywords:

 03.67.Ac;
 02.40.Sf;
 03.65.Vf;
 05.30.Pr;
 Quantum algorithms protocols and simulations;
 Manifolds and cell complexes;
 Phases: geometric;
 dynamic or topological;
 Fractional statistics systems;
 Quantum Physics
 EPrint:
 4 pages, 3 figures