Estimating Turaev-Viro three-manifold invariants is universal for quantum computation

Gorjan Alagic, Stephen P. Jordan, Robert König, Ben W. Reichardt
2010 Physical Review A. Atomic, Molecular, and Optical Physics  
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
more » ... idable 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.
doi:10.1103/physreva.82.040302 fatcat:lqcbgycsrrdxvktmmzeshjhgb4