# Discretization of SU(2) and the Orthogonal Group Using Icosahedral Symmetries and the Golden Numbers

## TL;DR

This paper explores discretizing the SU(2) group using icosahedral symmetries and golden ratios, resulting in a structured approach for approximating elements of SU(2) with potential applications in quantum computing.

## Contribution

It introduces a novel discretization of SU(2) based on icosahedral symmetries and golden numbers, providing an efficient method for approximating SU(2) elements.

## Key findings

- The constructed root system has a natural structure of SU(2, R).
- The reflection group H^∞ is of index 2 in O(4, R).
- Any SU(2) element can be approximated using five fixed reflections.

## Abstract

The vertices of the four dimensional $120$-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group $H_{4}$. There are two special coordinate representations of this root system in which they and their corresponding Coxeter groups involve only rational numbers and the golden ratio $\tau$. The two are related by the conjugation $\tau \mapsto\tau' = -1/\tau$. This paper investigates what happens when the two root systems are combined and the group generated by both versions of $H_{4}$ is allowed to operate on them. The result is a new, but infinite, `root system' $\Sigma$ which itself turns out to have a natural structure of the unitary group $SU(2,\mathcal R)$ over the ring $\mathcal R = \mathbb Z[\frac{1}{2},\tau]$ (called here golden numbers). Acting upon it is the naturally associated infinite reflection group $H^{\infty}$, which we prove is of index $2$ in the orthogonal group $O(4,\mathcal R)$. The paper makes extensive use of the quaternions over $\mathcal R$ and leads to highly structured discretized filtration of $SU(2)$. We use this to offer a simple and effective way to approximate any element of $SU(2)$ to any degree of accuracy required using the repeated actions of just five fixed reflections, a process that may find application in computational methods in quantum mechanics.