We discuss in details the role of Wigner 6j symbol as the basic building block unifying such different fields as state sum models for quantum geometry, topological quantum field theory, statistical lattice models and quantum computing. The apparent 2-fold nature of the 6j symbol displayed in quantum field theory and quantum computing -a quantum tetrahedron and a Computational gate- is shown to merge together in a unified quantum-computational SU(2)-state sum framework.