The Isomorphism of 3-Qubit Hadamards and $E_8$

Abstract

This paper presents several notable properties of the matrix $\mathbb{U}$ shown to be related to the isomorphism between $H_4$ and $E_8$. The most significant of these properties is that $\mathbb{U}$.$\mathbb{U}$ is to rank 8 matrices what the golden ratio is to numbers. That is to say, the difference between it and its inverse is the identity element, albeit with a twist. Specifically, $\mathbb{U}$.$\mathbb{U}$-$ (\mathbb{U}$.$\mathbb{U})^{-1}$ is the reverse identity matrix or standard involutory permutation matrix of rank 8. It has the same palindromic characteristic polynomial coefficients as the normalized 3-qubit Hadamard matrix with 8-bit binary basis states, which is known to be isomorphic to E8 through its (8,4) Hamming code.