  A quantum algorithm for approximating efficiently 3--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 in the sense of Wiesner and Crutchfield, where each basic
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 in Refs 2,3. 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
3-manifolds and connections with algorithmic questions that arise in geometry
and quantum gravity models are discussed.
