A quantum algorithm for approximating efficiently three-manifold topological invariants in the framework of SU}(2) Chern-Simons-Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided. The model of computation adopted is the q-deformed spin network model viewed as a quantum recognizer, where eachbasic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found by the authors. Thus all the significant quantities - partition functions and observables - of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite k. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and three-manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.