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Quantum geometry and quantum algorithms. (English) Zbl 1110.81157
Summary: Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the coloured Jones polynomial. The construction is based on the complete solution of the Chern-Simons topological quantum field theory and its connection to Wess-Zumino-Witten conformal field theory. The coloured Jones polynomial is expressed as the expectation value of the evolution of the $q$-deformed spin-network quantum automaton. A quantum circuit is constructed capable of simulating the automaton and hence of computing such an expectation value. The latter is efficiently approximated using a standard sampling procedure in quantum computation.
##### MSC:
 81T80 Simulation and numerical modelling (quantum field theory) 81P68 Quantum computation 81R50 Quantum groups and related algebraic methods in quantum theory 81Q60 Supersymmetry and quantum mechanics