The Ponzano-Regge model

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 twofold nature of the 6j symbol displayed in quantum field theory, quantum gravity 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. The Ponzano-Regge model represents historically the first successful model for discretized quantum gravity in dimension 3, and in this paper an overview of its applications is reported.