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.


81T80 Simulation and numerical modelling (quantum field theory) (MSC2010)
81P68 Quantum computation
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81Q60 Supersymmetry and quantum mechanics
Full Text: DOI arXiv