Spin-networks in the ZX-calculus
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Abstract
The ZX-calculus, and the variant we consider in this paper, the ZXH-calculus, are formal diagrammatic languages for qubit quantum computing. In this paper we will show that this language can also be used to describe SU(2) representation theory. To achieve this we first recall the definition of Yutsis diagrams, a standard graphical calculus used in quantum chemistry and quantum gravity which captures the main features of SU(2) representation theory, and we show precisely how Yutsis diagrams embed within Penrose’s binor calculus. We then subsume both of these calculi into the ZXH-calculus. In the process we show how the SU(2) invariance up to a phase of Wigner symbols is trivially provable in the ZXH-calculus. Additionally, we show how we can explicitly diagrammatically calculate 3 jm , 4 jm and 6 j symbols. It has long been thought that quantum gravity should be closely aligned to quantum information theory. Our results demonstrate 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.