Efficient quantum processing of 3–manifold topological invariants

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 [1], 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 [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. PACS: 03.67.Lx (Quantum Computation); 11.15.–q (Gauge field theories); 04.60.Kz (Lower dimensional models in Quantum Gravity); 02.10.Kn (Knot theory); 02.20.Uw (Quantum Groups) MSC: 81P68 (Quantum Computation and Cryptography); 57R56 (Topological Quantum Field Theories); 57M27 (Invariants of knots and 3–manifolds); 68Q15 (Complexity Classes) ————————————— (1) [email protected] (2) [email protected] (3) [email protected]