Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation

Details Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation

2010-03-03

arxiv.org/abs/1003.0923v1

Phys. Rev. A 82, 040302(R) (2010)

Physics

Quantum Physics

4 pages, 3 figures

Scientific paper

10.1103/PhysRevA.82.040302

The Turaev-Viro invariants are scalar topological invariants of compact, orientable 3-manifolds. We give a quantum algorithm for additively approximating Turaev-Viro invariants of a manifold presented by a Heegaard splitting. The algorithm is motivated by the relationship between topological quantum computers and (2+1)-D 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 Turaev-Viro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation. This establishes a novel relation between the task of distinguishing non-homeomorphic 3-manifolds and the power of a general quantum computer.

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