On some projective unitary qutrit gates

TL;DR

This paper explores topological quantum gates for qutrits derived from braiding six anyons in a specific topological quantum field theory, revealing new group structures and ancilla states for qubits.

Contribution

It introduces a new interpretation of certain braid groups in topological quantum computation and constructs novel ancilla states for qubits.

Findings

Identifies the Freedman group as a specific D-group in a new basis.

Provides a physical interpretation for Blichfeldfeld generators.

Constructs new ancilla states for qubits.

Abstract

As part of a protocol, we braid in a certain way six anyons of topological charges $222211$ in the Kauffman-Jones version of $SU(2)$ Chern-Simons theory at level $4$. The gate we obtain is a braid for the usual qutrit $2222$ but with respect to a different basis. With respect to that basis, the Freedman group of \cite{LEV} is identical to the $D$-group $D(18,1,1;2,1,1)$. We give a physical interpretation for each Blichfeld generator of the group $D(18,1,1;2,1,1)$. Inspired by these new techniques for the qutrit, we are able to make new ancillas, namely $\frac{1}{\sqrt{2}}(|1>\,+|3>)$ and $\frac{1}{\sqrt{2}}(|1>\,-|3>)$, for the qubit $1221$.