Quantum geometry and quantum algorithms

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