Spin-networks began life as an attempt by Penrose to see if continuous geometry could be generated from coupling spins together: Geometry from angular momentum. Seen through the lens of tensor-network theory these resemble a network where each tensor is SU(2) invariant and the contraction indexes now range over irreducible representations of SU(2) (read spins).  They remained somewhat a technical novelty however till Smolin and Rovelli realised that the state-space of quantised space itself can be given a basis by Hilbert spaces decomposed in this form. Still more recently Marzuoli and Rasetti speculated that spin-networks can be viewed as a kind of quantum computation a principle which has subsequently been restricted to what is known as permutational quantum computing which in turn has led to non-trivial results relating group structure and (super)-exponential quantum computing speed-up. Finally these objects are presently being investigated by the company Xanadu as a construction suitable for the quantum mechanical analogue of convolutional neural networks.

In this talk the spin-networks will be introduced and connections will be drawn between the disparate fields of their applications with the hopes of spurring interest in the application and generalisation of these concepts while hinting at a broader concept of 'harmonic tensor networks' or 'G-networks' formed of group invariant tensors contracted over spaces of irreducible representations of groups.