A fermionic code related to the exceptional group $E_8$

TL;DR

This paper explores a fermionic code encoding three qubits into sixteen Majorana modes, revealing its connection to the exceptional group E8 through a novel glueing procedure involving stabilizer codes and Fock space embeddings.

Contribution

It introduces a new formalism for constructing fermionic codes from stabilizer codes using a glueing technique linked to the E8 group, expanding understanding of fermionic code structures.

Findings

The fermionic code cannot be derived from a single qubit stabilizer code.

It can be constructed from two stabilizer codes via a glueing procedure.

The code subspace corresponds to a Cartan subspace of E8.

Abstract

In this paper we study the Hamming-like fermionic code encoding three-qubits into sixteen Majorana modes recently introduced by Hastings. We show that although this fermionic code cannot be obtained from a single qubit stabilizer code via the usual procedure however, it can be obtained from two, qubit stabilizer ones via a glueing procedure combining both single and double occupancy embeddings of qubits into fermionic Fock space. This technique identifies the code subspace as a Cartan subspace of the largest exceptional group $E_8$. In arriving at these results we develop a general formalism for implementing this glueing procedure via the use of intertwiners between different subsectors of the Fock space realizing embedded qubits.