Playing with Bob's toys to quantize gravity

The ZX-calculus is a formal diagrammatic languages for qubit quantum computing.

We show that it can also be used to describe SU(2) representation theory at the basis of the spin-network representation of quantum states of space. We recall the definition of Yutsis diagrams, a standard graphical calculus used in quantum chemistry and quantum gravity, and we show how it embeds within Penrose's binor calculus. Then we show how we can explicitly diagrammatically calculate 3jm, 4jm and 6j symbols with ZX-calculus. It has long been thought that quantum gravity should be closely aligned to quantum information theory. In this paper, we present a way in which this connection can be made exact, by writing the spin-networks of loop quantum gravity (LQG) in the ZX-diagrammatic language of quantum computation.