The ZX-calculus as a Language for Topological Quantum Computation

TL;DR

This paper uses the ZX-calculus to represent and analyze topological quantum computation models, providing new graphical methods and solutions for Fibonacci and Ising anyons, and connecting algebraic equations to ZX rules.

Contribution

It introduces a ZX-calculus framework for topological quantum computation, including new solutions and graphical derivations for Fibonacci and Ising models.

Findings

Graphical derivations of braid equations for Fibonacci and Ising anyons

A new exact solution of the P-rule involving the Golden ratio

A graphical procedure for simulating Fibonacci anyon braids

Abstract

Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for 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 important rule in the complete ZX-calculus known as the P-rule, which enables one to interchange the phase gates defined with respect to complementary bases. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as $\pi/2$ Z- and X-phase gates, 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. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.