Abstract
Certain quantum topological invariants of three manifolds can be written in the form of the Gaussian sum. It is shown that such topological invariants can be approximated efficiently by a quantum computer. The invariants discussed here are obtained as a partition function of the gauge theory on three manifolds with various gauge groups. Our algorithms are applicable to Abelian and finite gauge groups and to some classes of non-Abelian gauge groups. These invariants can be directly estimated by the nuclear magnetic resonance (NMR) technique used for evaluating the Gaussian sum.