anyon-pyzx

PyZX is a package written based on the ZX calculus. It represents circuits in the ZX-calculus and simplifies them based on a list of rewrite rules. For further information, see 1. This notebook shows the simplifying rules and braid generators of two topological quantum computing models, i.e. Fibonacci and Ising. It is an accompanying code with 2.

In this paper, we rectify confusion around a category describing an anyonic theory and a category describing topological quantum computation. We show that the latter is a subcategory of Hilb. We represent elements of the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang-Baxter equation is directly connected to an instance of the P-rule of the ZX-calculus. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as $\pi/2$ phase rotations, whereas in the Fibonacci case we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons.