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 in the sense of , 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 "Spin networks, quantum automata and link invariants" and "An efficient quantum algorithm for colored Jones polynomials." 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.
"Efficient quantum processing of three–manifold topological invariants." Adv. Theor. Math. Phys. 13 (6) 1601 - 1652, December 2009.